Free: Contests & Raffles.
The 308 would be a good choice, or whichever rifle you have that is most accurate. Coyotes aren't worth anything, so it shouldn't matter much how big a hole you make.
Some of the last few stands I've done, I havnt had any yotes come in. I am getting crows to come see what all the screaming is. Is this a good thing? Bad? Or just normal? For crows to come in.