deerhunter34,
I believe you have already made up your mind on a .223Rem caliber, and I know that this story of mine is not the exact same as using a .22mag on coyote. However, I believe the point of the story is very relative.
Many years ago I was squirrel hunting with a Ruger MarkII pistol. While sitting in a favorite spot, extremely still, eyes closed & about to doze off, a couple squirrels started barking within 20 yards. I could see one which was about 15 to 20 feet up in a tree on a knot where the tree had changed its direction of growth. I couldn't see what they were barking at, but I knew it wasn't me. All of the sudden a "different" furball appeared on the knot. It was a bobcat. The squirrel escaped. I sat there amazed for too long and he(it) slowly came down and went through a small gully. When he came back into sight at about 20 yards, I put the sights behind his shoulder and squeezed the trigger. Well, he screeched, jumped, and bolted through the ground cover. I waited about thirty minutes and then began searching for blood & hair. Well, I found NOTHING. I attempted to track him, i.e. I walked in the direction he ran. I was quite anxious trying to trail a wounded cat with a .22LR handgun. I never found it.
I was young when that huge disappointment happened and I share the story to prevent others from having the same disappointment. The same goes for deer hunting. You want to make sure you use a proper caliber that will kill quickly and leave a decent blood trail IF the particular animal you shoot is "tougher" than most. I'm not saying go overboard (.50BMG for coyote), but try to get something that will produce the desired results. I believe the .223Rem will do it for you.