I throw everything but the kitchen sink in the run. I agree, if at all possible you might want to expand your run. In the coop about all they want to do is lay eggs and sleep. You are certainly on the right track with free access between coop and run.....so why not make that run as comfortable and spacious as possible?
On the floor of the run, toss down some wood chips. They love to dig through them. And toss in a generous amount of dry leaves, complete with the little twigs that get raked up with them. Those twigs add air spaces essential for breaking down the litter in the run and help with drainage, as well as keeping the leaves from packing down. The more variety of leaf shapes and sizes the better. You can add some dry grass clippings, pine needles, garden refuse, A bit of straw provides a little insulation on the floor of the run, with it's hollow stems, but not too much of that - just a bit from time to time. When I first started I used nothing but bales of pine shavings. Nice, but they just don't break down well. The same with large amounts of straw. Stir it up every so often, but when you toss a little treat in there you'll find that the chickens are more efficient at that process than we are! They can scratch and stir that surface up like nobody's business! The goal is a nice, soft deep litter that doesn't have to be changed weekly (I do partial changes once year - or even less at times) Some peat moss tossed on top also helps insulate and keep things drier.
Good luck!!
