A solid bolt action .30-06. With bullet weights from 90 to 220 grns, you can download it for squirrels or take the full power loads for deer, elk, moose or bear. Using 150gr to 165gr bullets, it makes an outstanding 500 to 1000 yrd sniper round. Plus, you can walk into any back woods supply store and buy or barter for .30-06 ammo.
As to using a .22LR for game getting, yes it can be done. But the range must be short and the shot very, very carefully placed. This is for the larger sized game like deer, etc. I personally would not want to try and take an elk or moose with it. All it would do to a bear is p**s it off.
Now a 9mm or .40S&W pistol/carbine combo is also a valid option here. Again, if the range is kept short, like under 50 yards, and the shots placed very carefully.
My $.02. YOMV.