I'm not sure it matters if you shoot deer after the season vs during season as it works ok the same in the end imo
If the plan is to harvest primarily does, it can make a significant difference when they are harvested, with an earlier harvest being overall more beneficial, because:
1. Early in the fall, button bucks are more likely to be significantly smaller than their moms, so harvesting only big does will make harvesting a button less likely,
2, Harvesting does early leaves more mast for the survivors. This can be significant in areas where mast is the major form of nutrition in the late fall/early winter.
3, Harvesting does early leads to a much more favorable Buck/doe ration pre-rut, making for a stronger & more visible rut.
4. Allowing does to be harvested early takes advantage of the enthusiasm of some hunters to hunt deer, an advantage you don't have late in the season.
5. Allowing does to be harvested early can spread out the demand for deer processing. Otherwise, the glut is during modern firearm season in Late November. This can also be better for Hunters for the Hungry, as most of the HTFH deer currently come into the processors at the same time as the processor's money deer.