I agree the 22 LR doesn't have the power overall to make a quick kill. I have dispatched a lot of coyotes while trapping and have been using a 17 cal HMR, it does a good job with usually one shot (I know its all about placement) but still a little light if you want that knock down power at 100 yards out. Like GF said, 223 or equivalent would be best choice.
I know some guys that drive yotes and use buckshot, would be nice to have with on them closer shots.
Buckshot will work, your effective range is probably around 40 yards give or take on your gun. A single pellet would kill a yote well beyond that if it connected but your chance of putting a quick end to it drops rapidly past 40. Hound guys will use them, if your calling usually one guy with shotgun/buck shot and one with 22 centerfire rifle. If I use buckshot I prefer 4 buck.
I have killed them with a 22 rifle, not recommended unless you have a close standing head shot. Best option is a .223/5.56 or any 22 centerfire which I prefer to easily cover most field & fence line giving you a chance before they are on to you. 06 will work just fine if that is all you have, yes its over kill see if you can find some 150 gr loads that shoot good. A lot of years I used a .257Rbts with a 75gr varmit pill that worked nicely. I just like the quick follow up with the MSR 5.56 platform.
My best luck is with snares.
Now that deer hunting is over in my area been setting up cables near bait sets. Have gotten two already. Pretty thick this year. Will expand my cables out further in the next week and then start hunting them. Shot one during archery and saw another with mange but could not get a shot on him.
HOLY COW! idk if its legal but I've been watching that pool 7 webcam on the Mississippi
for a couple months now, and when its been iced over there are coyotes all over out there running around all day, you see 2 or 3 at a time, its weird. easy pickins if you could get out there.