Let the carcass hang cool overnight.
• to stiffen a sort of rigor mortis, and so the meat becomes more tender again, and
• for cooling off: this is a kind of maturation process to get tender meat muscles.
During these processes, there are found a variety of changes taking place in the cells of the muscle and to the cross-links (consisting of collagen) between the muscle tissue. The moisture that forms and is released is called drip. The much more remains in the flesh, the much more tender it is. But you have quite to give the opportunity and the time to instill without it drying completely.

Much larger the animal, much longer it may hang for we find it tasty and tender. In cold storage this can be up to 3 days (for wildlife), but since the production should go ahead there are many other techniques (like electrical stimulation) to accelerate these processes.
A pig, a week; a beef is even hang up to 3 months, obviously cooled.

This is also called maturing the meat.

This is one of the reasons why there was slaughtered in late autumn, on days when it was cold enough to have no burden of flies who are out to lay their eggs in the flesh. You can see the clusters of whitish dots properly. But if there are many, it is an endless tinkering to remove them. So better avoid them.

Don’t confuse hanging with becoming nobly (faisandage) of game. For this, people let game, especially fowl, sure pheasants hang with their guts 10 to 14 days at 8 to 10°C. (Even a month!) Partly due to the intestinal micro flora present during the maturation phase, a rotting process is in motion. This makes the meat more tender and gets a strong ”taste nobility” (haut gout,or high game). The meat is then marinated for 24 to 48 hours in a mixture of wine, aromatic herbs and vegetables, which is oil an acid. Probably to camouflage the strong flavor. Because of bacterial and toxic risks this practice is rarely used today. But specialist still prefere that pheasants, grouse, partridges and geese will have a conditioning time of 3 to 7 days, depending on how old they were.

Hang the carcass high enough overnight so that other sweet tooths (dog, cat, mouse, rat) cannot reach it.