I only used a 22 lr on a coyote once, and I'd never do it again. The shot was perfect broadside and I chose the shoulder shot, more habit from big game hunting than any other reason. Well it nocked him down but he got up and ran off. I tracked him for over a mile and never caught up with him. About a month or so later I trapped what I believe to be the same coyote with the bullet logged in his shoulder. The reason why I think it is the same dog was because I caught him approximately 50 yards from where I had shot him, and he was back to using his normal cycle again.
Go for the .22 mag if you have to use a rimfire. The .22 lr doesn't have what it takes, saying that, I'd have killed him if taken a head shot and hit him in the brain. Lesson learned.