From 1985 to 2015, I filled 21 mule deer tags with a Ruger M77RL Ultralight in the pipsqueak .250 Savage and I never had to shoot one more than once.
Here's what the terminal ballistics of my .250 Savage handload topped with 100 grain Nosler Partition bullets did from my rifle (Distance / Velocity / Energy)
M / 2620 / 1524
100 / 2352 / 1229
200 / 2100 / 980
300 / 1865 / 772
As weak as that metaphorical sauce is, if you hit a mule deer in the heart with that, it doesn't do anything but die.
In the here and now, I can get similar terminal ballistics from 5.56 NATO handloads topped with 77 grain Sierra Tipped MatchKing bullets. For the sake of comparison, here's what that load does out of my home-assembled 20" barreled AR-15 A4 style rifle:
M / 2852 / 1391
100 / 2634 / 1186
200 / 2426 / 1006
300 / 2227 / 848
If I can fill tags from one-shot kills with pipsqueak little shits like these, you'll have no trouble doing likewise with anything from .243 Winchester to .308 Winchester or 6mm Remington to 7 x 57 Mauser or .25-'06 to .30-'06 or any kind of 7mm or 300 Magnum.