I can NOT recommend anything rimfire, for coyotes. Coyotes are tough customers and can take a lot of punishment/bullets and not die. At that point all you've done is educate it for the next person. Rest assured, you will not see that coyote again, as it may not leave the area, but will become nocturnal.
A centerfire lever action rifle is good for quick follow-up shots, without having to resort to the more expensive semi-auto actions. Shot placement is MOST important, as there are people that have killed coyotes with rimfire rifles...and it can be done, but there's no room for error. Coyotes are not known for offering the best shot angles.
Depending on your wants/needs, you could purchase a Mosin-Nagant 7.62x54R, for about $160 and some soft pointed ammo, which would certainly get the job done. Then you could use the rifle for other large game hunting.
Good luck with your choices!
If God didn't want us to eat animals, he wouldn't have made them out of meat.