the 7mm really is a step up from the class or cartridges that begins with the .243...
If you look at the spectrum of .308win cartridges as a whole, it's not as clear cut as "a step up" in increasing recoil and increasing game weight potential, but most people don't consider the entire spectrum, and only consider the front runners - .243win, 7-08rem, and 308win - in that consideration, there's a baby brother who must have been a "whoops" with two older brothers closer in age.
But if a guy looks at the entirety of the spectrum, you don't see quite as much of a stair step between the players.
It's not surprising, bullets will follow a similar profile, so you'll see a fixed spectrum of bullet weights for a respective caliber, and when you put the same case behind them, comparison is straight forward. But the .308win case is really capable of any game in North America, and managing bullets from 35grn up to 250grn...
But if you pick any two neighbors on this list, not skipping a few calibers, you won't find a significant gap. Pick cartridges spread by a few calibers below and you'll see a bigger gap. Such is the nature of these things.
22 CHeetah and 22-243 are really meant for the lighter bullets, typically loaded with 60 and lighter, but it could be loaded 35-90grns if an appropriate barrel were selected. As to be expected, these aren't elk rifles, and strain for whitetail sized game, but really shine with small game up to coyotes. Of course, these are both wildcats, but the 22-243 does have a strong following.
The 243win as the 6mm entry manages 55's up to 115's, given appropriate twist. These can be a bit much for small game, can be kinda nasty on coyotes for a hide hunter, and are much more efficient and effective on whitetails.
I'm not sure many folks will have ever heard of the .25 Souper, and likely for good reason - it runs 75-115, 120grn bullets, and largely doesn't do anything the .243win can't. A wildcat proposition here too. Effective, but less readily sourced and not really worth the hassle over .243win, although I think it would make a fine midrange speedgoat cartridge, very capable on whitetails as well.
The 260rem slides up to the 100-140grn ballpark, closing the door on ANY question of lacking power anyone might have had with the .243win. Until recent years, a guy didn't hear much about the .260, but it has enjoyed a resurgence (probably a bad choice of words, as it's more popular now than it ever was), as the x47 and Creedmoor have bolstered a precision long range light cartridge market. Nobody is recommending it for Elk, but it's a hammer on deer, and certainly effective for elk when deployed.
Ackley lists a 270-08 in his manual, but if memory serves it was meant for the 120's and 130's, and it's about as obscure as a cartridge can get, considering the common nature of its brass and caliber. This one COULD likely bridge up to 150's, but I've not heard of anyone doing it.
Stepping up to the 7-08 offers access to long 160's, but seldom do folks actually shoot anything so heavy in it. Typically hanging out in the 120-147grn ballpark, but could be loaded up to 160's. Still not highly recommended for elk, there's never any argument it is not so capable, and I have occasionally heard mention of its use for moose. This tends to be about the line where people cut off their recommendation for youth rifles. In a light rifle, a proper youth model which could be carried by a 12yr old, the recoil could be too stiff for a child of such size. As it could also be for a 140-147grn, or 160grn 260rem. On the extreme side for coyotes, it's typically only used at extreme ranges or by guys not collecting fur.
The venerable 308win, of course, is best fed with 150-180grn bullets, and the recoil crosses over the line for many youth shooters in light, short rifles. It also crosses firmly into the "too much" realm for coyotes and small game, often even touted as such for pronghorn.
338 Federal is a little power hammer of a cartridge. It runs out of juice for the 250+ pills, so it doesn't garner as much favor as larger case cousins for moose and bear, but it's still quite capable for anything in North America. Recoil is commensurate with the 180-225 pills it's running. Nobody is handing a 338fed to their 12yr old daughter, and rightfully so. It's one of the better behaved 33's, but it's still running a 200grn pill.
The .358 Remington is one of those grand old cartridges which earned the right to ride the "positive side of obsolecence." The .338 federal likely wouldn't have garnered the favor it has if there weren't such a sparse market for 35cal bullets, and such a good supply of 338's. It dutifully manages the 250's, but like many of the .308 family, it runs out of mag length to load the super heavies, and runs out of horsepower for anything but short range work. A guy can be sure plenty of eastern bruins have fallen to .358win's, but I can also be sure you won't catch a millennial carrying one.
Pick your target game and you'll likely find 3 from the list which are well suited - the smaller caliber of the 3 will run lower recoil, giving you a few options to fit your preference.