The beauty of a deep litter pack is when it gets a good soaking the healthy, spongy soils under the DL will absorb the fluid quickly, the nutrients really drain into the soils when that happens and the worms and bugs respond accordingly. This creates even more openings in the soil where the fluid can be absorbed as the worms tunnel upward from beneath. You'll find the top of the DL will look dry in just a couple of days of the soaking as this good drainage and absorption takes place. What you'll also find is your chickens doing a massive excavation of the DL to get the bounty of bugs and worms, thus aerating the layers even more, hastening the drying of the litter.
If you have sustained rains and snows and it stays wet all the time, that would be no different than if you had barren, packed down and slick dirt/mud in the run or if you had sand, so any floor you have at that point will stay wet....the only difference is the DL will be a healthy wet~no mud, no stinking, wet, fly covered poop slick and no stinking, sodden sand~ and the other two, not so much.
Take a look at Rosemarie's run pics...pure heaven for chickens! It's large and she uses long pine needles in the run. She found out just what DL does for the soil when she was working on the run and moved that DL for a bit....underneath the soil was darker, loamier than the soil right next to it outside the pen.
If I were you, in the winter months I'd place tarps on your hoops to keep out most of the snow and on the one side of your run that gets the most rain and snow, I'd place another tarp. This gives your birds a place to get exercise out of the coop and your litter will still get moisture but it won't be thick with snow when the chooks would like to go outside. You'll really want that for them come winter...that coop can get mighty small when they all have to stay indoors all the time.