If you are careful on a nasty field dress, the worst thing that is affected are the tenderloins on the inside. I have never done much with the ribs and the flanks. The skin should protect the rests if you leave it covering the inside of the hind quarters. I quarter my deer at our camp, and have the meat on ice for several days.
Between a nasty gut job, and skinning, wash hands, or change rubber gloves.
Most all of the meat can stay segregated from anything that might worry you
C. Davis