Whatever kind of shot it took to down the deer, just dress it out quickly, and if you can, wipe the inside of the cavity out, do it. then HEAD HOME, IF it was a dirty gut shot, Hose the cavity out (with cold water only) as soon as you get home A.S.A.P.
Note: I Always hose out the cavity regardless of where or how it was shot.
IN most cases you have to hang the deer overnight because the deer has to be reported at a weigh station/check-In station/ reporting station or whatever your state calls it...
IF the outside temperature is 40 degrees or lower you're all set to just hang the deer head down Overnight and the meat will be fine.. IF it's going to be warmer than 40 degrees GO to the store and buy a bag or 2 of Ice and place the bags in the cavity of the animal and that should be all that needs to be done for an overnight hang...
the Butcher will determine if any of the meat is bad... In most cases The meat won't be affected at all...