I know how to hunt Coyotes.
I have a father who is an avid and very proficient predator caller. We went out last Sunday and I was able to get 4 coyotes with one getting away. I was using my new Rem 700 Vs in .308 with Speer 125 gr. TNT HP. Let me tell you, even using the 125's, the first coyote I shot at about 200-250 yards out and he by no means died instantly. When I found him over the hill, he was about 15 yards from where I shot him, still alive with a very large hole in his side. The next shot was enough. Still, with that big explosive bullet, that coyote still didn't go down. Later in the day I one shot, one killed the next two with the .308. Interestingly, my next shot was on a coyote facing me. The bullet entered in his chest and there was no exit wound! His insides must have been complete mush! The next coyote died right away too from a near front on shot, but from I was shooting from above him. It opened up his entire belly and spilt his guts out everywhere. That .308 will take the coyotes down if you hit them right. And if you don't hit them right, they still go down, but not out unfortunately. The last coyote went down from a 12 gauge at about 15 yards using I believe a mix of some T buck and #4's. Quite the lethal coyote mix. Slugs for coyotes? Why not just use a rifle?
And so, if you plan on using a .22 rimfire for shooting coyotes, plan on doing a lot of searching for the coyote. My dad used to shoot his trapped coyotes in the head with a .22 short, but that was at a range of about 2 yards and with careful aim.
Chang, get a 10/22 for just the joy of shooting. Then build up as you get more experience. But if you are just going to be shooting at coyotes that you just happen to see running around, you will need a scoped rifle of a decent varminting caliber. Otherwise, you are just educating the coyotes.
And finally, I have shot a coyote once with my M1 Carbine. It was a quick drive by shooting on a farm road, so I was limited to my FMJ. It stopped him, but I had to go out and finish him off. If you use the M1 Carbine, make sure you get some HP or soft points and make sure it is within at least 100 yards. Otherwise don't waste your ammo.