After having killed well over a thousand coyotes for both pelts and for predator control over a 50 year period, I've taken them with everything from 22RF to 50BMG including many with 17-222 & 17 Remington. Using any 17 cal in my opinion usually has little pelt damage but a much higher animal loss. Even the 22 hornet won't anchor a dog with consistency, the 223 with 50-55 GR super explosive projectiles is more consistent and gives a bit better range limit. A 6mm with fmj does very well too but shoot thru ricochets can be a concern! In the hay days of high pelt prices I used a 222 with great success on both coyotes and bobcats but calling at nite a 12ga with 4buck still rains out to close to 100 yards. I'm lucky because of land holdings in 2 states that allow nite hunting for predators that until nite/thermal scopes it what I used.
The one coyote I shot with a 50BMG using an Amax at 600yds had a quarter size hole in and out, it was loads of fun but not my idea of a coyote gun!
My recommendation is shoot what is accurate and hold the range to get the best chance of a clean kill. The person who taught me to hunt coyotes was a pure pelt hunter and he killed far more than I have all with a 22mag and rarely took a shot over
100yds.
Gain calling and techniques that work and you will discover a hunting sport that is very satisfying!
Hunt hard, be safe and have fun!
Doc
gun)