Also, when skinning, cleaning, and quartering/deboning...
Skin and clean as soon as possible to let heat out of meat. If you cannot debone and it's hot, cut to expose bone to let more heat out for quicker cooling.
The knife that touches hair should not touch inside the animal. Any urine, gland secretions, etc. on the hair will be transferred to the meat and taint it.
Do around /near glands last. If you nick or cut a gland, the knife must be washed or set aside and a new knife used. Otherwise you'll transfer that to the meat and taint it.
The knife that skins should not cut meat. Again, chance of nicking a gland and tainting the meat.
A knife used for cleaning that comes in contact with offal or other gastric fluids should also be washed or set aside so as to not transfer to meat when cutting muscle tissue.
Use of game bags will keep meat cleaner for packing and transport to a cooler.
Your game handling in the field has as big an influence in the quality of your table fare, as the hanging, if not more. (Honestly, I think more)