I thought I would chim on this one. As previous postings have mentnioed, get it clean and cool as quickly as possible. If the animal is gut shot, you split the stomach or bladder, then I try and rinse the carcass, but I am no more then a mile or so from pottable water. Here in South Central Michigan, the deer mostly live on corn, beans, grains & hay. Same thing the cows are eating, so I have not noticed a large improvement in taste by letting the animal hang. Might be also due to targeting 1.5+ year old does.
I butcher my own and it is a family affair complete with cold draft beer, double grind the bruger, vaccuum packaging, football on tv, some of the meat going directly to the frying pan, etc... I really enjoy cutting it up and knowing how it is taken care of.
Regardless, if it is warm or freezing, I field dress, then hang, skin it and boneless quarter the animal. I learned this technique through McClendon Meats videos on YouTube and originally a link with the near same technique on this forum. I then throw the pieces in Cabela's game bags, put them in a large cooler and fill the top with ice. I open the cooler drain and make sure it is on an angle. The temp in the cooler is between 34-40 degrees and as the ice melts, the water cleans the meat as well. Really helps when we are way up North. All you have to do is keep replacing with fresh ice.
Never liked the thought of having a deer travel in the back of a truck or hanging at camp for a couple days, then bring it to the processor after running it through highways of exhaust, dirt, salt on the roads, freeze & thaw cycles, only to have the processor hang it for another couple days. I really notcied a difference in quality & taste by doing it myself.