Gotta disagree here. A .22 mag will do a coyote just fine with proper shot placement. Even a .22 Long Rifle will take out a coyote with a head shot at close range.

With a .22 mag, use 40 grain or heavier hollowpoints and take head shots. Provided you are a good shot and your rifle is accurate enough, you should be able to shoot out to 75 yards or more. A centerfire is better for coyote of course, but at closer ranges your .22 mag will work just fine.