I use last fall's leaves as litter on the floor of my coop. Neighbors dutifully bag up their leaves and set them out by the curb for me -- whether or not they know they're doing it for me.
(I also compost leaves directly. I use the leaves that people collected through mowing for compost, and those collected via raking for chicken coop litter. Yes, I start out with a huge pile of bags of leaves in November. I'm just about through using all of them. I might have to scrounge for something to use in October until the leaves start coming again.)
I also cut wild grass and alfalfa from the nearby fields several times a week and scatter it throughout the chicken run. Chickens eat the heads of the grass, and they gobble up the alfalfa. Usually a thin mat of grass and alfalfa stalks build up in the run. And that layer helps keep the mud down during rainy periods.
Once a week I rake out the coop, and then rake that whole pile across the chicken run, and it collects all the stuff -- including poops. It all gets tossed on the compost pile, and the worms do their thing.
Know what helps compost piles decompose more rapidly? Used coffee grounds. they're high in nitrogen, very fine, and they usually come very moistened, so it adds needed moisture to the process. At least in Colorado Springs, Starbucks packages up their used grounds (also with used tea bags and banana peels -- all good for compost) and leaves them for gardeners to take. I read somewhere that 40% by weight of a coffee shop's garbage is coffee grounds, so if they can reduce their trash by giving it away, it's a win for them, and a win for me!
When it's time to turn over the garden soil in November, pretty much the whole compost pile (except for perhaps the last 3-4 weeks worth of added material) is ready to spread and mix in. By springtime planting, the soil is nicely fortified and ready to grow!