In which period would native americans have been most likely to practice hunting and gathering

Native Americans have been hunterers and gatherers up to the period when growing crops became available to them/when they discovered growing crops is a thing that could be done. The reason for this is that when you've started growing crops, you can safely stay on one spot and have a sufficient supply of food for a longer time period - this wasn't available before. However, even with the coming of agriculture, the still hunted and gathered as it was an important aspect of their life. 

South central region