Okay, so we talked about what scout rifles are, and why .308 was generally the "universal" scout rifle chambering, but as in all things, sometimes the general utility option isn't appropriate.
The primary TECHNICAL (as opposed to logistical) reasons for choosing .308 in a scout rifle are it's length, and the fact that it is in the maximum power range for most people to be able to make fast snapshots and followup shots.
Other lighter chamberings may be appropriate (some have suggested .243, 6.5 grendel, 6.8spc, and other intermediate calibers), but for a "true" scout rifle, the chambering must be as powerful as one can conveniently make a snapshot and followup shot with. For most folks, that means a .308, a .270, a .30-06, or something in that power class. It generally excludes the more powerful magnums unless one is a VERY experienced rifleman.
Actually, even for a very experienced rifleman, the heavier magnums can be a handful. I personally consider anything over .300 win mag to be in the "Only if my life depends on it" range for making a quick snapshot and followup, and I'm a recoil rhino. It's not painful to me(though it will certainly bruise you without a thick shooters pad); but it IS difficult to control that rapidly. The .375 IS just flat out painful for more than one shot (the first shot is shocking, but the pain doesnt come for a few seconds) in rapid succession.
That generally excludes these chamberings from a scout rifle, unless there is a specific need for them (on an elephant or rhino hunt for example).
Also excluding these chamberings is their length; or more particularly the length of action and barrel required for them.
Magnum rounds are large cases loaded with large quantities of slower burning powder (generally anyway); and therefore require a longer action to feed them, and a longer barrel to provide good performance. A short barrel will create a huge muzzle blast with increased noise and flash, increased recoil, increased muzzle deviation, and signifciantly reduced velocity over the typical 24-26" magnum length barrel.
On should note, this is why a "true" scout rifle isn't chambered in .270 or .30-06; both of which are highly availble, and both of which are adequate for Elk given proper load selection (heavy partition or solid bullet, heavy charge). Unfortunately, the OAL for a .30-06 (and .270) can be up to about 85mm (or longer with non standard bullets), so both also require a long action with most bullet selections. The .308 at about 74mm max oal (72mm is standard) is just about the longest round you can fit into a short action rifle and still have clearance for different bullet weights and seating depths etc... Some short actions CAN in theory accomodate up to an 80mm OAL, but 74-76mm is generally the limit to ensure function with all ammo - The original short actions were based on either 75mm, 80mm or 3" which is just about 76mm.
18-19" is also about the shortest barrel length you want for a .30-06 class cartridge; or a .308 for that matter. In fact either will have a significant muzzle blast when compared to the typical 20" or 22" barrels for hunting guns in those chamberings; and a 200fps or more loss in muzzle velocity. The .30-06 for example was originally concieved for a 28" barrel, which was the standard full military rifle length at the time (though it was originally issued in a 23" barreled rifle as a compromise).
Overall this shows why .308 is considered the "proper" general scout rifle cartridge, especially in continental north america. Its short, fast to handle, relatively easy to shoot, and mostly powerful enough.
Okay, so that's continental North America; where the biggest threat you're likely to see is a pissed off blackie or MAYBE a brownie, and while a .308 isn't an ideal choice for those, it's better than nothing; what about more dangerous game areas?
Move the venue to Alaska, or Africa, and priorities change a bit. Now you are looking for adequate performance on large dangerous game; without vastly increasing the size of your weapon.
Generally speaking that means you want a big, heavy, relatively slow bullet in a relatively short case (again, handiness is important). It means transitioning to a long action (at least 12mm over a short action) but not to a long magnum action (which is at least 6mm longer than a long action and may be up to 24mm longer - any longer than that is .50bmg category).
A note on action sizes. I've talked about short, long, and long magnum action sizes, and a little bit about what they mean, but lets go into a bit more detail. The original short bolt actions that we are talking about now were researched around the end of the 19th and beginning of the 20th century by the english and the germans; but because of World War 1 most of that developement was put off. The developemnt resumed between the wars, and viable short action cartridges were produced in the 20's. Then a worldwide depression and WW2 put off the further growth of the short actions popularity until the early 1950s when they just exploded (especially with the .308 winchester/7.62x51nato chambering).
In the original actions made for short cartridges, the english were looking at actions to accomodate 3" oal (76mm), and the germans 75mm or 80mm oal.
The original long actions were mostly English and German medium to big game hunting or miltiary rifles, of the 1880s to early turn of the century. These designs were licensed, copied, folded, spindled, and mutilated across the globe to come up with the majority of chamberings and rifles that became common in the US, UK, and in Germany up until WW2 (and by extension in africa).
The english long rifle cartridges were based on 3.5" or 3.75" (89mm or 95mm) max OAL and the long nitro/long magnums on 4", 4.25", or 4.5" oals (102mm, 108mm, or 114mm). There were also hunting rifles with action lengths supporting up to 5" (127mm) OAL, though the longest hunting cartridges I can think of are in the 110mm oal range.
The german rifles were similar, though their long actions were generally based on 90 or 100mm, and the long magnums on 110mm maximum OAL (the biggest I can think of is the 9.8x84, which would have had about a 105mm OAL - I've never seen one, only read snippets).
A selection of the designs I'm thinking of would include the Mauser models 92 through 98, the 1903 springfield (a direct mauser copy), the 1895, 1904, 1907, 1916, pattern 14/1917, 1924 mark III and 1939 mark IV Enfields, and the various Steyer-Mannlicher actions (Mauser influenced but quite different). Most "modern" hunting rifles (which were actually mostly designed in the '30s, and refined in the '50s), are a direct evolution of the designs mentioned here, Blaser and Weatherby being notable exceptions (both are all new actions developed in the '80s, and the '40s respectively). In particular the two most popular hunting rifles in the U.S. are direct evolutions of Mauser (Winchester Model 70), and Enfield (Remington 700 - really both mauser AND enfield) designs.
One should note that we didn't ALWAYS copy the English and Germans. The third most popular hunting rifle in America is an American classic; the Winchester lever gun (mostly the '94 model, and it's many copies, immitators, refinements and evolutions). We were hunting and scouting with our short action lever guns 20 years before the euros were developing their long action battle rifles.
The lever gun is short, light, handy, fast to to get into action, fast to followup with, and only slightly slower to reload than a good boltie. Honestly it makes a great scout rifle; and it is the original American scout rifle. Other than giving up a slight edge to the bolt action in reliability and accuracy, the only real issue with a lever is a limited cartridge selection because of the mechanics of feeding. Basically the chamberings need to be relatively short, and use flat pointed bullets (the new hornady lever revolution loads notwithstanding).
When you move into the long action ranges, your chambering choices open up. Remember though, ammo availability is still a critical concern, as is snapshot ability.
Generally speaking the big and slow (again relatively) philoshophy is the way to go here, since you are looking to have managable recoil with massive stopping power on dangerous game. You really want to be thinking 8mm (.318) and above.
This is where some of the older cartridge choices come into play like 8x57 or 8x62, 9.3x57, 9.3x62 or 9x64, and the .375 to .415 class (like the medium bore rigbys) non magnum calibers.
9.3x62 is considered one of the finest medium game chamberings by experienced African and European hunters; and for many years was one of the most common chamberings on the continent. It is adequate for thin skinned dangerous game, and thicker skinned but less agressive game. It isn't really adequate if you are going to be in an area where there are Rhino, Elephant, or Buffalo (or grizzly bear in north America), though yes, many of each have been taken with the round; and it's better than nothing (or anything lighter for that matter, which would be about as good as nothing - but the 9x64 and 9.3x64 are both decent choices).
The .376 steyer is ballisticaly similar to the 9.3 cartridges (it's 9.55x60), was specifically developed for scout rifles; and is the longest and hardest hitting non-magnum chambering that would be reasonable for the short action, short barreled scout (it JUST fits in the longest short actions). Unfortunately it's a very uncommon chambering, and expensive.
If your venue is the Rockies or Alaska, or even Africa if you don't mind not having a local source of ammunition, don't discount the .45-70. Yes it is an old, fat, slow, blackpowder based cartridge (the designation means .45 caliber, 70 grains of black powder), but so are many of the "fine english doubles" so to speak.
One of the great things about .45-70 though is it's SHORT. The OAL of the .45-70 is standard at 2.55" (65mm); it's medium length case with a very short (outside the case anyway) bullet, and guns chambered for the .45-70 can be correspondingly short. There are a great many lever action and single shot rifles in the chambering that are quite light, short, and handy, even with long barrels; because the cartridge itself is so compact.
The .45-70 can be downloaded to "cowboy" load levels, say 275gr at 1650 feet per second (1650ft lbs), or uploaded to 300gr at 2450fps (4000 ftlbs)... even all the way up to 550gr heavy solids at 1750fps (3400ftlbs, but a 550gr solid is NASTY). Of course not all guns are safe, or will even funtion with all those loads.
The .45-70 can be anything from a pussycat to a monster when it comes to handling and recoil; especially since it's chambered in relatively light rifles. The cowboy loads for it kick about the same as a .357 magnum in the hand, or lighter than a .30-06 on the shoulder. The maximum loads kick something fierce, but also deliver massive impact energy. I can shoot cowboy loads in an 1895 Marlin all day long, but I don't know how often I'd like to shoot a 4000 ftlb load in a six and a half pound gun.
The only downside to the chambering, other than it's lack of easy availability outside the Americas; is that it is generally only chambered in lever actions, and single shots. If one chooses a lever action scout rifle (as MANY Alaskan and Canadian professional hunters and guides do), then this is no detriment, but for many the bolt action is preferred; both personally, and for the mechanical reasons discussed in previous posts. I know of no currently manufactured bolt action sporting rifles chambered in .45-70.
All that said, I'd have no problem trusting my life to a Marlin big bore guide gun, XLR, or 1895ss (especially if massaged by wild west guns - personally I want the Alaskan Co-Pilot or Guide somethin fierce. Either one would do.)
Going back to long actions, you can also (barely) Squeeze some of the older non belted nitro express calibers into a long action without going to the long magnum action, depending on exactly which action (and exactly which bullet). Your case size limit on a long action is usually something in the 64mm-66mm range with a long bullet, but if the chambering is a short bullet (or very long necked) design, some case lengths up to 68mm or even 70mm may fit.
If you ignore the big and slow dictum, everything up to about the .360 magnum class (non belted) will fit into a long action if it's a short case design; but anything in the .375 and above class really needs the long magnum action, as do basically all the traditional belted magnums. Unfortunately, from a barrel under 24" a lot of their performance potential will be wasted.
Oh and for the metric heads among us, the .375 H&H is also known as the 9.5x72 (there is an earlier non belted 9.5x72 and a non belted 9.3x74 and 9.5x74 as well), and it has a 92mm nominal OAL. Long actions can really only accomodate a 90mm oal or so.
There are a few notable exceptions to that .360 and below rule though. For example, the .338 lapua is a chambering gaining popularity in the tactical shooting world, and it is a long case design which very definitely requires a long magnum action. At 8.6x71mm and with a VERY long bullet design (for ballistic efficiency), it's oal is around 94mm.
Again, anything much bigger and you get into .50bmg territory, which is 12.7x99 with a max oal of around 140mm (it varies depending on the chamber reamer and bullet profile). Yes there are extra long magnum actions in between (like the various .460 to .510 range wildcats - most of which are modern versions of old nitro or blackpowder chamberings; and which range up to 110mm or even 115mm OAL), but they arent exactly scout rifle material.
Though some lunatics have made bolties that big for African hunting, most everyone who wants to play with a ridiculous big bore hunting round does it with an english style double; and nobody uses optics on these guns unless they plan on replacing their forward mounted scope (too much recoil for rear mounts off hand) after every shot.
Now where did I put my coconuts...