I have a Ruger 10/22 with a 3x9 40mm scope on it zeroed for 75 yards. Although you can use the .22lr on coyote, I would consider 75 yards to be a max distance. Beyond that, it will not have the energy to do the job. Regardless of what range, you'll end up tracking if you use the .22lr.

The .308 will give you the range and energy to anchor the yote without tracking. Sure, the pelt damage will be greater. Check the local laws first, but you can get a 147 grain FMJ (full metal jacket) bullet that will eliminate the pelt damage problem and it still provides a solid take down.