No, I will never process a deer the next day as that is when it is the toughest.
University of Wyoming did a study on aging venison. Antelope to moose. 24 hours after the kill is when the meat is at it's toughest then slowly tenderizes.
Antelope 3-5 days
deer 5-7 days
elk 7-10 days
moose up to 14 days
Of course this all depends on proper care of the venison on good temp control. They suggested 34-37 degrees, warmer speeds the process, colder slows the process.
You can also cut up immediately, but it slowly gets tougher for the first 24 hours, then reverses.
In a perfect world, I have my venison skinned, and quartered, crusted and into heavy cotton games bags and into the spare refrigerator within 4 hours. Loins and tenderloins vacuum packed and into the freezer.
The beauty of aging is that you have a three day window to process and that's is good thing with everyone's busy schedule in today's world.
If you are pressed for time, turn the fridge temp down to 33, if you have to get it done, turn the temp up 38.
To me the quality of venison in both taste and texture is vastly improved with proper aging.
I have never been able to find the study again to save a copy for my records, I will try to do another search and see if it is back on the internet.
Edited to correct temps from my memory lapse.