Hang the field-dressed carcass with the hide on (insulation against bacteria) at consistent 33-40°F. Avoid temperature swings. Process when meat reaches a deep ruby color and a slight tackiness on the cut surface.
For warm-weather hunts where hanging isn't safe, bone the deer out and dry-age the cuts in a refrigerator at 35°F for 3-5 days. State disease risk pages cover food safety guidance.