Professionally speaking, i am a hunting professional chef, you need to gut as quickly as possible, but time and temp are everything.
If its cold, you must gut quickly before it freezes. If its unseasonably warm, you need to get the core carcass temp down to about 34-44 F a quickly as possible, otherwise the meat will go rancid.
Letting them lay overnight might be okay depending on what time they actually expire and what the over night temp is.
With all that said, meat in general is dry aged under climate controlled conditions at about 45-50 F under ultraviolet light to stop mold and fungus growth. Meat is wet aged in cryovac plastic about 40-46 degrees. Meat aging is really just controlled rotting, it needs to rot a little to taste good. Meat that is processed before it gets through rigor mortis and softens is called green cut, and has a very coppery flavor to meat(comes from blood being to fresh, like when you bite your tongue. I find that many hunter's family's don't like venison, because its not process/aged correctly. If the weather is good (50 in day and 30 at night) I let my deer hang in root cellar, for 4-7 days depending on size of carcass.
Good Luck and Good Eating!
