Gut it where it drops, haul it out and hang it long enough for the rigor to pass, then butcher away
There is no reason to hang wild game like a side of beef (where you are letting the fat break down to help tenderize the meat). Most antelope, deer, elk, etc. don't have anywhere near the amount of fat it would take to help tenderize the meat. The longer you hang the carcass, the more meat dries out and becomes exposed to bugs and other contaminates, and the greater the chance of it starting to go bad and getting thrown in the trash.