I think the original question was about how to correct it once the damage is done. I'm not sure there is a good solution to that problem. Most of the suggestions are preventative in nature, and they're right on!
I field dress immediately, right where the animal falls. If there's a water source nearby, I dunk the carcass to get the cooling process started as quickly as possible. When I bring the animal out of the woods, I skin it immediately and hose it down thoroughly. I know some people will disagree with that step, but it has worked the best for me. Get all the hair off, and all the excess blood inside the body cavity. If it's not cool enough to hang outside, which it rarely is in MS, I remove the backstraps and quarter it, and put all of the meat in a cooler on a thick bed of ice, arranged so that no water from melting ice will pool on the meat. I cover the meat with more ice, tilt the cooler and open the drain plug so that the meat NEVER sits in water. The old refrigerator idea is even better. In a week or 10 days I cut it up, trimming it closely and removing all the fat. If I want summer sausage or anything other than steaks and burger, that's when I take it to a reputable processor, one you KNOW will take good care of YOUR meat.
To save processing time and freezer space, instead of cutting the steaks while I'm processing, I freeze the big muscles whole, and cut them up when I'm ready to use them. That gives me the option of making roasts, steaks, stew meat, or whatever I am hungry for at the time. The meat holds up better in the freezer in bigger pieces, too.
I bought a 1000 watt grinder from Northern Hydraulics that will grind anything I put in it without bogging down. I remove the majority of the "silver skin" sinew, but a little bit left on won't taint the meat, and speeds up processing time considerably. Getting it off is the most time-consuming part of processing. I do get ALL the fat off- that's where the bad taste is concentrated.
The grinder also has sausage-making attachments for links and brats and such, but I haven't tried that yet.