Originally Posted by Red84
Id say use your .308, you owe it to the animal, even if it is a coyote it still deserves a quick death. I have seen a few taken between 30-50 yards with a .17 and it does fine, anything further, go with your .308.
If you are going to keep/sell the pelts, you want as small a round as possible. I use a .223, but it'll knock a bigger hole than I like - too big of a hole to sew up. My hunting buddy has a .22-250 Howa, and it's a perfect gun for dawgs. A lung shot passes right through, but with a lot of energy - they'll drop right there. The hole is pretty small too.
On the .17HMR: I agree with Red. Up close, especially if you are hunting the smaller Western doggies, but be sure you do good shot placement so he dies quick. Stay ethical. It's not fair to the dog to use an underpowered round.