I ouwld hate to go into theelk mountains undergunned. My last (only only so far) elk was killed with a 280 remington chambered model 70. I din't notice the recoil them but since then I have shot the same gun and it is noticably less than a .300 win mag (the most popular elk cartridge). But I have read about many elk being killed with the verable 6.5x55 sweedish. This is a great caertidge and not too nasty to shoot. Itis already extremely accurate. You might also consider the 7x57 mauser and the 270. Both kick less than a 30 06. But don't forget the .308. All of these would be good out to say 200 or 250 yards.
Browning makes a BOSS system which is a muzzle break that also can enhance accuracy by tuning the vibrations of the barrel. These are (or were available on Winchester model 70 and are still available on Browning A bolts. Most manufacturers like savage and Remington makle guns with muzzle breaks but seems to me that they are mostly on large calibers.
If money is not too tight, you can get your rifle and take it to a gun smith and have a muzzle break installed. Some reports I have read claim 20-30% less felt recoil.
Anyway, good luck
okcmco