One of my problems in replanting potatoes is that they have to break dormancy before they will sprout. I've never been successful just putting them into the ground. I even tried ordering potatoes from the Maine Potato Lady for late season planting and she sent me some that should have been planting in the spring. I only got a couple to sprout out of two different varieties and they did not produce anything. Like you Steve, I occasionally get volunteers from missed potatoes. Never in time for that year. If they showed up the next spring in a place I could not leave them I'd often transplant them to where I wanted them. They are a relative of tomatoes and transplant pretty well plus I took as much undisturbed dirt with them as I could. All that was in Arkansas.
According to the Louisiana extension services planting calendar I can plant them down here from mid-August to mid-September. If I can find local seed potatoes then I may try some. But only if I can find local seed potatoes so dormancy is not a problem.