We built a deer cooler to prevent the worry of hunting in warmer weather. Its on here if you search for it. If you are getting it processed by a butcher bet it out of the wood ASAP and get two bags of ice. One in the chest cavity near the front shoulders and one in the rear cavity and between the hams. If processing yourself you need to break it down to cooler size as quick as you can.
If a deer sits over night in 50 degree weather it will be good. 60's iffy and 70's I highly doubt it. Meat will not rot in a couple of hours even at 80 degrees but over night can be pushing it. You have to weigh the possibilities on your own and what you think could happen. I quit an hour before dark in the early season as I am not leaving a deer out overnight and consider it being wasteful.