Good question.
The deer population had been all but wiped out in most areas of the state sometime before Indian removal. Of course there were probably a few left in the most remote parts of the state. They had practically been hunted to extinction for over 100 years during the deer skin trade era. I think they were already pretty scarce by the time of the 1813-14 Creek War. The lack of wild game probably contributed to the starvation and defeat of the Red Sticks, especially after they had slaughtered all the domesticated European animals (cattle, pigs, chickens) they could find in or near their villages.
My guess would be turkey, quail, and squirrels were the most popular for hunting in 1860.