I'm no meat processing expert, but I can say from experience that if the meat is hanging in temps much higher than 36-38 degrees, it is basically rotting. I just finished cutting up my whitetail that has been hanging in temps around 33-35 degrees for 5 days. It was like butter. I do not like hanging my deer without the hide on. Some may disagree, but I don't like the dried out shell that develops on the exposed meat. My 2 cents. I would let the temperature dictate how long to hang. If you can keep it at 34-35 degrees, 5-6 days is not a problem. Any higher temps and you better think about cutting him up quicker.
Course, I live in Saskatchewan, so most years I have to THAW the deer out before I cut them up.....