We can shoot "varmints" year round (you don't even need a hunting license to shoot feral hog any longer) BUT...
deer fall into a different category. You can get special permits to take additional deer but the permits are only valid during hunting season, you have to take a bunch of pictures to prove you have excess deer, blah blah blah...., submit requests, wait for some "wildlife biologist" determine how many special permits you can be given.
5 years ago, we went through the hassle and then received 3 doe permits... 3 days before the season ended. The next year we went through the process again, only to be given only 2 doe permits a week before the season ended. 3 years ago, we couldn't even get the damn biologist (there is only 1 guy who makes the determinations and he is over 200 miles away and doesn't give a damn about farmers) to return phone calls. Last year, we didn't even try to get any permits.
This year, we can shoot up to 2 does during the first 2 week of the season. Although we only have 150 acres on that tract, the brush and trees are so thick on the 50 acres of creek and pastures, with hills from washes, we set up 5 stands/blinds safely. We could probably set up 3 more sites easily. My brother, B-n-L, nephew and great-nephew have their own stands, in addition to mine. All are set up for downward shots. My great-nephew's is the closest at less than 200 yards from mine, the trees are so thick multiple full belts of 50 cal wouldn't punch through.
The problem we have is that taking out 5 to 7 (if we get 2 more today) will not alleviate the problem of too damn many deer. The current population would not be naturally sustainable if not for the fact that they are chowing down on crops.