I agree with The Bear. You need to take enough gun to put the thing down hard and not have it take off bleeding and make it even more dangerous.
Hunting Coyotes is about as close to hunting the ultimate prey as you can get. They are very crafty, they are very wary and what works today ( if you are lucky ) probably won't work again. They are smart predators and they are almost arrogant in their taunting of you about it.
If a coyote paw could flip you off, I am pretty sure I would have seen it happen to me by now.
My second time out, I was shooting with my .308, and there was a coyote pack that was out at about 250 or 275 yards. We had been trying to call them in, but only one would show itself.
Arrogant little bastard seemed to know where we were and was on this slight rise, just pacing back and forth, then would drop down after about 3 or 4 paces, just as I was developing a rthym of where he was going to be.
This went on for about 25 or 30 minutes and it was frustrating as all hell. See him pop up, head on, not a great shot, then turn sideways and pace a bit, then back down the other side of the rise. Talk about scope fatigue.
I had my choice of no shot, taking a moving shot, or trying to get closer. Thinking I was one with the Universe, I tried to creep closer while my buddy kept position and was watching with his binocs.
I made it about 12 yards before the coyote just up and left. I am pretty sure he farted in my general direction when he left too.
I love the hunt when going after coyotes. They are definitely a tough shot. Don't get discouraged if you don't get one, and don't get to cocky if you do. Your fortune will change either way... LOL