Retired from whitetail hunting, but I experimented with this a good bit during the rut, rainy days as well as on draw hunts on federal land where deer hunting wasn't otherwise permitted. On the cases from the later examples, I'd often kill deer at the end of the morning shift, midday, and just before the evening shift hours as deer would move around the flood of pressure.
For the rut, since mid morning and mid day chasing is so common, I tried doing 9AM to dark sits for awhile as that captured the prime time hours for rut behavior and allowed me to get more rest for back to back full days. That really didn't jive with me though and I settled in on hunting early morning until mid afternoon, around 3 PM. Getting out of the woods early, eating dinner at normal hours and going to bed early worked better for me and allowed me to stay motivated for putting as much time in a tree as possible.
Rainy days are another good time to be in the woods mid day. If its not raining too hard or if you have multiple fronts moving through, you'll often see midday movement. I killed some big bucks outside of the rut on all day sits in the rain.
That being said, you might not follow my advice as I completely burned myself out on whitetail hunting.