There are lots of different caliber Rifles out there to choose from...thing is You need to decide if Your saving the fur/pelts or not,a larger caliber Rifle will do more damage and if Your wanting to save the hides or even give them away to a Trapper it would be better to use a Rifle with smaller grain bullet or a smaller caliber Rifle.Now I have been using my Browing Bolt-action .243 Rifle with 100 grain Hornady bullets..if I was to drop down to a 60,50,40 or even 35 grain bullet it would reach out further and be easier on the Predator/Varmints fur.I opted to leave my .243 as a back-up Rifle and bought a CZ-USA model 527 "Varmint" in a .204 caliber using a Hornady 40 bullet or might even go with a 35 grain bullet or I heard a 39 grain Sierra Blitz King bullet is more accurate because of the way the bullet is shaped.
Here are a few more Caliber Rifles I feel are just as good for hunting Coyotes,Bobcats,Foxes and so on...!
22-250 Here in Missouri this caliber can be used for Predators,Varmints and Deer.
.243 also for Predators/Varmints and Deer
.223 again Predators/Varmints and Deer
.220 Swift Predators/Varmints (not sure about Deer?)
.204 Ruger Predators/Varmints
.22 Hornet Predators/Varmints
.17 Remington Fireball Predator/Varmint
.17 HMR Predator/Varmints and small game
There's probably other calibers out there I've left out but those are several that I have looked at and considered using to Hunt with.