I have to throw in a recommendation for one of my favorite low-recoil rounds, the .260 Remington. If you're a handloader, this cartridge is fantastic. You can load 80-100 grain frangible bullets for varmints and coyotes, 100 or 125 grain Nosler Partitions and 129 grain Hornady bullets for deer, and you can go all the way up to 160 grains for bigger game. I would use it for elk or black bear with 140 grain and heavier bullets. It wouldn't be my first choice for that game, but a 260 would do the job if I did my part.
If you don't reload, go with the 25-06, I suppose. The .243 isn't as flexible as the slightly larger rounds, since bullet weights and sectional density in available loads are lighter than a lot of folks want.
Frankly, I don't recommend calibers less than 6.5mm/.264" for medium game and up, although I'm well aware of what the 24 and 25 calibers can do. The 26s just happen to be the smallest caliber rounds with good sectional density bullets widely available, which means good penetration is easier to get, even with non-premium bullets. My kids will likely get 6.5x55mm or 260 Remington rifles when they begin deer hunting.
There's no reason to beat your shoulder with a WSM, not with your intended use.