Get it cooled down ASAP, let it age. My deer doesn't head to the butcher shop for at least a week minimum. Your best beef is aged 28 days, not sure why guys are scared to let their deer hang awhile
Because, among other reasons, Venison does not have the same enzymes in it as Beef, and hanging essentially makes for a bigger pile of trim on the butcher shop floor, for the same exact cost to cut.
I prefer to eat mine, rather than chuck it away.
Further to that, you can age beef wet or dry, but freezing it pretty much does a lot more for breaking down the cell structure. A decent dry aged steak has never seen a freezer.
Now, to the OP, gut where it lands.
If it's cool enough, leaving the hide on won't hurt, may even keep the carcass from drying out too much. If it's warm enough out to be not wearing a jacket (half retarded teen agers don't count!) the hide should really come off unless you are within a few minutes of a cold storage site that you can use. If you are that worried abut contamination, don't hunt the waste dumps!
In all seriousness, skin it on a clean tarp next to the car, drop in onto another clean tarp for the trip home.
When you gut, take a look around. Other than the tenderloins, there is bugger all really, that you can splash anything on to, that cannot be washed or trimmed away before processing.
Even a badly gut shot deer can be cleaned out with a hose before processing (but preferably as soon as possible after gutting, while still wet).