I've used .17 Rem, .204, 222, .223, 22-250, 220 Swift, 22/243 and 22 Middlestead (which is just a pain in the butt version of a 22/243).Then there's the 243 which is a great killer of coyotes you don't want to skin and not much else. After that pick a big game caliber and I've likely killed some yotes with it.
The 17 is a waste of time, the 204 isn't any easier on pelts than a 22/250 at close range and drops off in killing power at longer range, the 222 and 223 are good for the pelt collectors that keep the range short, the 22/250 is great but you better be careful about bullet selection (hint 52 grain Berger Varmint match and believe it or not white-box Winchester) if you're skinning. 22/243 tears 'em up; 243 is great for coyotes you aren't going to skin and it just gets worse from there.