I've taken a bunch of backyard/barnyard coyotes, skunks, raccoons, cats, possums, porkies, etc. using either a Ruger PCC 9mm or 77/357. I haven't worried much about wasting a handful of range ammo over the years, and the bigger bullets definitely work better than .22lr on critters that you aren't eating or skinning...plus they're more fun than .22 for this purpose. The PCC works just fine, but the 77/357 is much more accurate and I usually have more appropriate bullet choices for it so that's what I tend to use.
But for actual hunting, I have carried a Rossi Ranch Hand in .44mag for deer a few times without firing a shot. I swapped that for a Henry .44 Mare's Leg a few years back, installed the full-size stock and an aperture sight, and really hope to take a deer with it soon. It is plenty accurate, it's easy to find hunting-appropriate bullets for the .44, and I honestly can't imagine why I would want to use the .357 instead...and not even considering the 9mm.