Actually I pile it up, and they scatter it out, breaking it down, by spring it will be very nice for mulching my garden and any weed seed will have turned into eggs. When I bring in fresh hay, mine will eat some of it.
I know you have sand in your run, but you might consider putting the hay on top of that, then when it snows just flip it, so that the hay is on top, and your chickens will come right out into the run. Do make sure you have wind protection in your run, you probably do, I am just mentioning it. The hay will absorb a lot of poo, and you can stir it up, and keep your sand clean and fresh for next summer.
If I think it is going to snow, I pile it up, cause it is easier to flip on top of the snow, but otherwise, my hens are happy scratching and spreading it out, gives them exercise.
This time of year, I have to keep them penned up in the run, nearly 24/7 cause the day is so short, dark when I get home, and the predators are so hungry, they are too brave. Stiring up the hay, or taking it out, and putting in new, keeps things cleaner. Chickens are such creatures of habit, last summer, they were out the moment I opened the gate, and a bit put out if they were not let out. Last Saturday, I was cleaning, and left the gate open, and nearly had to chase them out into the big outdoors, they were so used to being confined!
MrsK