When you shoot a deer you'll notice that it is limber immediately after the kill. Then in a short while it will stiffen (rigor mortis). Processing while the deer is in rigor is a mistake, from what I've heard. I always hang until the rigor is out of the carcass. If it is too warm, I add ice.

When it's time to eat the venison, I take the meat out of the freezer a couple days ahead and let it thaw in the fridge. I put it in a colander with a bowl under it, so it will not sit in the blood that comes out as it thaws.

As long as I've butchered my own (about 27 years), this has worked for me.