Compost is slow at this time of year, when its too cold for it to begin to heat up and decompose. When (if ever?) it gets warmer, you will need to make a fourth bin and start turning the accumulated stuff out into this one, layering in green material such as grass clippings or cut nettles ( lots of minerals in nettles). This process will introduce the necessary oxygen and will mix the old stuff with new material that will heat up rapidly and thus spread through the heap. To make good compost as fast as possible, you do need to keep on turning it, so it all gets mixed and aerated, so you need a spare bin for circulating the other bins into. You also need to keep the heap covered, I use the kind of bubble wrap with giant bubbles you can get from garden centres by the metre, held down with a couple of planks. This is better than the old carpet which is sometimes recommended as carpet gets very heavy and horrid when wet, and also grows harmful mould spores which are inhaled when you disturb it. The best sort of bin is large and cubic, not those silly little plastic Dalek things. Get 4 old pallets, tie them together with nylon rope, and line the inside with plastic sheet so stuff doesn't leak out of the sides and it all stays nice and cosy. When the bin is full, just undo the rope and you will have a neat cube of lovely earthy compost all ready to dig out. The bigger the bin, the better it will heat up, and the faster you will get good compost. (Or get 7 pallets and make 2 bins.) chicken muck makes brilliant compost and will get very hot if mixed with the right stuff and allowed to get air all through it. Very satisfying.
I agree about the hay, I think this is pretty useless both in the coop where it gets really messy, and in the compost, where it takes forever to decompose and may have grass seeds in it which germinate unless your heap is really hot. Better to use Aubiose or similar such as Cosibed, which is easy to clean out, warm and cosy, and decomposes easily into lovely compost.