I don't want to kill coons, foxes or cats. Nor do I like to make an animal suffer for any prolonged amount of time. GW
With today's offset modern traps, 90+ % of catches result in no foot damage whatsoever in my experience. I've released coons, possums, and dogs completely unharmed. In all my trapping, I've only had 2 animals with any damage, both with traps I don't like to use anymore. One was a coyote that barely caught toes, and one coon back leg. I don't like that either and have changed some things to hopefully prevent that in the future.
But animals suffer much longer when hunters shoot and wound them but they don't die from it. Pick your poison. Whether you hunt or trap, the best you can do is keep getting better at it so that you minimize suffering.
Even the DNR in some states uses foot hold traps to study, tag, and release foxes and other such animals.
But, of course, it's totally up to you what you decide, I was just saying that trapping will be more effective.
I'm not the authority on tactics for hunting coyotes, but if you post a new thread asking SPECIFIC questions as to techniques someone will be able to help you.