You sure can! It's where it makes the most sense. Pile it deep and keep something along the base of your run to contain it so the birds can't kick it back out again. It will not only help with the smell and mud, but will help discourage flies in the summer. Since using a composting deep litter system, I haven't seen a fly....and that's been 4 yrs now. And you don't have to clean it out at all, just keep adding to it.I can do this in the run? Our girls area is dirt and right now it's basically mud with all the rain we're getting. If I pile a layer of leaves, wood chips, pine needles (which we have plenty of all 3 of those in our yard) it'll help with the wet mucky smell out there?
Eventually it may develop such a thick layer of compost or soil that you may want to remove some for garden use each year but you'll never have to remove old bedding material...it will get digested into the mass and turn to compost/soil.
Building it deep will insure the excess moisture wicks into the bottom layers and the top will stay mildly damp during wet weather but you'll no longer have puddles and mud. Any time you see such as that, you don't have your deep litter deep enough, so you'll need to add more.
I'd store up all the leaves you can right now so you'll have enough litter materials to take you through spring and summer of next year. Folks are bagging them for you in every little town and just throwing them away. The pickings are ripe right now!