For reasons that Colockum spelled out, yes, the red light is better for 'your' vision recovery as dawn progresses. But as others have stated, the deer don't care. You're likely busted by the noise and the wind at night just as fast as during daylight. If you want a chance at some better night vision, as soon as you wake up, put an eye patch over your dominant eye. When you get to the trail or even part way through, using your red light, you'll be able to see much better when you uncover the eye.
Warning- You'll have to manage your balance and depth perception when walking the trail/driving a bit more in earnest when your eye is covered.
Me? I use my white light on low or nearly no light at all well before I need to be on stand. By the time dawn comes around, my eyes are recovered pretty well.
-Steve