In years gone by I used a number of the bigger .22 centerfire cartridges, including the .22-.250, and .220 Swift, after a long stint with the .222 Rem. Eventually I moved up to a larger caliber cartridge as they provide better wind bucking abilities and hit with more authority out at longer distances.
My wife now shoots a .243 and I use a .257 Weatherby. I can poke them a long ways away and if an opportunity presents itself at a wolf, I still have plenty of punch.
Big thing is bullet choice, as western coyotes are one of the few bright lights in the fur market these days and I really do not want to shred them. I wait until the hides are prime before I get after them, with the exception of the odd yote that starts to hang around the livestock or tries to lure our dogs out of the yard.
If you are just shooting them to thin down the population and the hides are not an issue, bullet choice is not an issue.