I use both 22 mag and 17 hmr every once in a while on predators. Carry one or the other all the time for sod puppies, beaver and other critters. I love my 17s and consider them one of my top 3 favorite rounds of all time. It won't have an issue killing a coyote with the right shot out to 100 yards. I wouldn't go much over that though due to energy loss. I shoot either a CZ452 or tikka T1x suppressed in the 17HMR and have a Bergara and a few Marlin bolts I shoot 22 mag in. I use a heavier bullet in the 22 mag with a 30 cal rifle suppressor on it and it will hit a bit harder at 100. I also wouldn't go much over 100 with it either.
Really can't go wrong with either caliber for what you are doing. Ammo is easy to find and cost roughly the same a box of 50($15+/-). I would recommend going with a good bolt action for either one over a semi. The ones I mentioned are very smooth and quiet shooting and loading. I highly recommend a suppressor for both and outfitting with a decent scope. All mine are topped with Zeiss fine cross haired scopes because the rifles are all very capable of great accuracy! Many beaver head shots have been placed and sod puppies have been successful eradicated out to 100 yards with them
If you are more fox and smaller coyote I would probably lean towards 17HMR. Great fur saver for those closer shots. Shoot for heart.