I say hide on if you're certain the carcass will cool quickly and you have a way of propping open the cavity. When my uncles and I used to stack up the deer back in the good old days (what, 8 years ago?) we'd leave the hides on and do that. Even with a few in the bed of a truck, they'd cool quickly. Plus the hide kept the meat clean while we were driving around. I use chest spreaders because they're pretty user friendly and seem to give a better hold than a stick.
Otherwise I get the hide off as fast as possible. It's way easier to skin when the deer is somewhat warm, and that will facilitate cooling, too. My ideal method of butchering is as such:
-- Shoot deer
-- Go home, hang deer, skin immediately
-- Strip out inner loins
-- Let hang over night (assuming it's cool enough)
-- Have a hearty breakfast with ample coffee the next morning, then spend the day butchering
-- Feel satisfied that I can provide and process meat for my family
Rinse and repeat.