To each his own, but I fall into OKRattler's court on this one. I expend an inordinate amount of effort to finish the job when I knock any critter over and he gets up. I will not give up until satisfied that there is absolutely no hope of finding it.
That's just me, and far be it from me to judge anyone who does not share my thoughts on the matter. However, realizing that coyotes have no mercy on their prey, I don't lose a lot of sleep over those I have been unable to find. Having said that, I do feel that I am obligated to do my best to put them out of their misery ASAP when I screw up.
I used to spend three days a week at the ranches I hunted and always went back the next day, if still at that ranch, looking for birds in an effort to close the book on a bad chapter one way or the other.
I once spent a full afternoon literally crawling through a really thick brush mott looking for a bobcat I hit that subsequently jumped into the thicket. Ended up losing my wrist watch somewhere in there when either the band or the pin broke.
Went back next day and retraced my route as best I could to no avail (found neither the cat or the watch). Went back the following week for another 3 day hunt and was drawn to the thicket as it was one of my favorite cat spots. Smelled the cat and had been within 15' of it the previous week.
Unfortunately, try as I might, couldn't smell the watch but in this case, the rancher's bounty on cats bought me another watch with a little change left over.
Regards,
hm