The harder the bullet, the less the damage. Use a bullet constructed for deer hunting and it'll produce limited damage on deer and coyotes. When I was hunting back home in college, I had about a 5mile drive out through our property roads out to my favorite deer hunting spot, so I'd often stop along the way home and pick off a squirrel or two with my .30-06 with ballistic silvertips. In general, my .223rem with hollow points does MORE damage to a squirrel, rabbit, or coyote than my .30-06 ever did. Harder bullets like a pointed soft point out of my .30-06 did LESS damage even than a .223 with hollow points. A .22lr or .22mag is definitely the least damaging, but hard bullets out of big bore rifles will also limit the damage.

One of my good friends and coworkers wanted to sight in his new Winchester SX-AR this spring, and had brought out a few boxes of surplus ball ammo (Full Metal Jacket), and after sighting in, we had an opportunity to shoot some Prairie Dogs... My .223 with hollow points was making pink mist, and his .308win with FMJ's was making 30caliber holes through them... Moral of the story, it's all about the bullet...