The only 'varmints' I go after are rabbits. One shot, one kill ... head shots with an air rifle. But, I have made a few bad shots.
When I can follow-up on my own property, in privacy, then that is what I do. But, I'm not going to charge through the neighborhood looking for a wounded rabbit ... that would be idiotic, possibly cause a panic, and get me involved in some long conversations that I choose to avoid.
And, I feel very guilty whenever I have only wounded a rabbit, and not made a clean kill. It has caused me to (1) become a better shot, (2) learn to pass on the 'iffy' shots, and (3) pass on the long shots.
Silly examples, perhaps, but I think this has made me a better hunter. And, I have to live with myself ... my own ethics require the decency of a quick kill. I don't like hearing me lecture and swear at myself.
Take care. Regards from AZ