What you want to do is get the meat below 40 as soon as you can and hold it between freezing and 40 at least until the meat relaxes and preferably until the meat loses its elasticity.
It really depends on how far you want to go.
The faster you get it below 40 the better its good to go then but will be tougher if you don't wait for the meat to relax. If you want to go even further you can dry age which is a process of dehydrating some of the moisture/blood in the meat. This blood(I can't remember what its called) is what causes "gamey taste".
Once the excess moisture escapes the meat will break down and tenderize. When that happens the meat will no longer bounce back when you apply pressure and you are ready to butcher.
A lot of people soak the meat which actually works against you.
You can actually dry age in a regular refrigerator as long as you leave the meat in big pieces. Just put it on fridge racks uncovered until the meat loses elasticity.