Depends on whether you care about the pelts or are just shooting them for pest/predator control and don't care.

If you are interested in maintaining he pelts in sellable condition, then you might consider a .243 Win shooting FMJ or a non-explosive expanding bullet designed for deer sized game. Both bullets will go straight through a coyote, bot won't leave a fist sized hole on the exit side like a varmint bullet will. The downside is that a non-varmint bullet, perticularly the FMJ, is far more likely to ricochet, making backdrop a major consideration and shots around livestock problematic.

The .223Rem is a great caliber for coyote inside 250-300 yards, but with varmint bullets the pelt damage can be severe. With the .22-250 and .220 Swift the exit wounds can be downright spectacular (instant disembowelment is very possible), making a coyote very dead, very fast, at the expense of the pelt.

Mike