You are good to go under 50 degrees for sure. It depends on the internal temp of the deer as well. If you shot it that night I would get ice in the chest cavity and skin it before you hang it.
We have hung deer at 55 degrees over night but that was a bit stressful and that was only after the carcass cooled to 40-45 the day before.
We hang a venison quarter for 40 days at 40 degrees in a modified refrigerator which times out perfect for a Christmas roast.
I've hung them overnight up to about 50 or 55 degrees. I also hang them in a pole barn that stays cool during the day. I wouldn't hang them outside in the direct sun. If you have a good shady spot and can get ice in the chest cavity you should be good overnight into the next morning.
Always a tough call trying to decide if it's cool enough and how long is it safe to hang. I'm glad I don't have to worry about it anymore since I built my deer cooler.
Thanks Guys. Where I hunt in IL the nights don't cool down very well during the early bow season. This sometimes keeps me from hunting, but I decided I'd like to start getting out earlier. Cutting up a deer right away is an option, but I'd prefer to hang it over night, then butcher in the morning. I usually have some frozen ice jugs in the freezer, so I like the idea of stuffing ice in the cavity. Actually I like the idea of having a deer cooler even more, but that's probably not going to happen. Thanks! Nice deer by the way.