Since you've already wounded the game, its already in pain, is it not? Pushing the animal would only cause it more pain and suffering. If you failed to make a DRT killing shot initially allowing sufficient time for it to expire would be the best thing. If you have to finish it off, a bullet in the ear hole usually does the trick.
Personally I'm a big believer in the high shoulder shot where the spine passes through the blades and enough gun to punch that area out. Take out the spine and you don't track them much at all.