I've never hung birds this way, but have on more than one occasion let them sit outside overnight in the cool fall air.
RanchoD - I always age my deer. The fresh killed and butchered deer I've eaten in the past was the worst I've ever had. Extremely gamy, smelled nasty cooking and tough. I've had arguments with my brothers about this and one year they shot two deer out behind the house on the day we were gathering to butcher the other deer shot during the season. Despite my objections they skinned and butchered it right away, placing the wrapped meat in grocery bags on the cold garage floor. The next morning we came back to a lake of blood. The meat, not having gone through rigamortis, had bleed out during the night. What a mess. They had to re-wrap all the meat and I refused to take any of it.
If the temperature is right I'll leave them for a couple weeks, hide on. 20 years ago my dad's best friend and our lifelong hunting companion, had a heart attack while dragging a 12 point swamp buck he shot opening morning. That week the temps got into the low 40's during the day and around freezing at night. the deer hung for 10 days. He said that was the best venison he's ever eaten.