I suggest a dedicated long range varmint (read: heavy barrel) rifle for coyote and other small predator hunting (like lynx bobcat and wolverine). A smaller bullet does a lot less damage to the pelts, has lower recoil, and often cost less.
Consider:
.17 Remington Fireball
.22 Hornet
.220 Swift
.223 Remington
.22-250 Remington
.204 Ruger
Or if you dare .223 WSSM
Savage Arms makes some really cool varmint guns at a price anyone can afford. Remington varmint guns are incredible but almost twice as much as savage. Bowning’s varmint guns are limited; better off with a Remington at this price point. If money is not a factor look at the Kimber 84M Pro Varmint or LongMaster VT. The CZ 527 VARMINT LAMINATED is a very cool gun as well.
If you are going to kill a predator please do it the honor of taking its pelt. (my 2 cents only)