I've went the opposite direction in Calibers...I was using my Browning A-bolt Hunter in a .243 and went with the 2 following Calibers that take down the Coyotes but leave them much "Prettier" and fur friendly if You want to save the hide or have it mounted plus it's going to be much easier on the fur/hide of smaller Predators if a Fox or Bobcat comes in while in Season! ;)
CZ Model 527 "Varmint" (longer,heavier barrel) in .204 shooting Federal loaded 39 grain Sierra Blitz Kings.
Savage Axis XP Camo .223 Bolt action using Hornady 55 grain V-max bullets.
Future Rifle-Ruger American .22-250 Bolt action using Hornady V-max bullets or Federal Sierra Blitz Kings?
2nd Future Rifle-CZ Model 527 "American" .22 Hornet.
It's all about good shot placement and calling those Predators in close for a good shot!