My strategy has always been to strive for a clean shot and clean field dressing. Then make sure the meat is chilled as soon as possible. If it is warm weather, I quarter the deer and place in the freezer right after skinning and process the next day (no "aging" at all). If cold (under 40) I let it hang for a day or two and then process. Once, a guy shooting damage control deer gave me some that was harvested in hot summer weather and not properly cooled - it had a bad smell to it and I had to discard it. For sure, getting as much fat out also helps, as mentioned. Another factor is deer diet - deer from swampy forested areas with little or no hay, corn, or agricultural crops will have a wilder taste than deer from farming areas.
Since the whole CWD thing I have been deboning the meat and we either make stew, steaks, loin, or I have it mixed at a reputable butcher with quality beef. One would be very hard pressed to distinguish our deer meat from beef except that it is leaner and actually tastes better!
Yes, the clean shot is a great idea. Supposedly a clean shot prevents the release of adrenaline which people have always talked about as adding to the gamey taste. I wish I had a way to hang a deer in 40 degree weather. We dont really have a place to do something like that.