Been hunting white tail deer in the deep south for 50+ years now. Mostly in SE Al. My experience with a clear, bright moon lit night is that deer tend to move later in the morning. The big driver down here is the weather. Hot weather, they seem to move very little in the day light. The cooler, the better ... as far as deer moving in day light hours. So you may be being hit by a double whamy.
When it is hot, I tend to hunt in the coolness of the creek bottoms, along trails or near dropping acorns .... and stay as long as my rearend will take.