For what it's worth, I've whacked 20 or so incidental coyotes over the years while elk or deer hunting with a .300 Win and .340 WBY....and with stout bullets designed for elk out of those rounds at normal ranges, I have found significantly less pelt damage than I have with the more accepted hypervelocity varmint rounds and their normal bullets.
Totally agree with TI. A tough bullet put through the ribs and staying off the shoulder will save the hide.
Use what you have for now and if you really get the bug to keep hunting coyotes, you can buy an more appropriate caliber down the road.
Personally, the biggest mistake I see made when using a deer/elk rifle for coyotes is being zeroed at too high at 100 for a critter the size of a coyote. The commonly used zero of 2-3" high at 100 will have you over shooting many coyotes at 150ish, unless you remind yourself to hold low. That can be hard to do in the heat of the moment.