I've been using an open bottom coop with a deep litter system in the winter and in a paddock shift system for the spring summer and fall. Once the ground thaws in spring, I pile up the deep litter (poop, wood shavings, hay and straw) and let it compost. The coop goes up into the field where it's moved weekly.
So far we have used our own weedy hay and straw left over from two neighbors' straw bale construction for bedding. Deep litter in their traveling coop. No smell! Last year's hay went straight into the annual beds. We had parked the chickens in that prospective garden area for the winter and just scraped it out onto the ground before dragging the coop to the next location. No issues with burning foliage. Got to love that Bertha rotary plow. When I get to cleaning the coop in spring, the straw will go directly into a new sheet mulch bed. This year the hens are wintering at one end of the hoop house. I haven't decided what to do with the winter straw. Or how to get more if nobody builds another straw bale house. I haven't quite heard an alternative yet for bedding from the no-till champions. Since goats deposit >50% of their manure in their sleeping area, and the chickens do quite a lot on their roosts, it's a non-trivial problem.