Roe, I dont really notice the difference freshly shot or hung in the chiller for a few days.
Reds - easier to skin when warm, I do like to partially skin - leaving the skin attached at the legs so the skin will cover the animal but be easy to remove once hung - saves the meat drying....
Once hung for a few days skinning reds is tougher, but not impossible by any means - good way of staying fit,,,,
Fallow - hate the greasy things - best skinned chilled IMHO - or by someone else!