How cold does it actually get? I ask because there are a few cover crops that will actually survive and keep growing down to 10 degrees F... You could seed winter peas, or like LG said, rye. They would both grow right through your winter, I believe. The peas are "nitrogen fixers", so the will actually PUT nitrogen in the soil instead if taking it away, much like clovers but hardier. An even hardier perennial clover is actually alfalfa. That will come back every year and not die with frost.
IMO, you'll need a cover crop either now, or in spring, or the grass will grow right back up from the bottom of the sod.
This reminds me of permaculture. The trick there is wood. Rotting wood, then flip the grass did upside down on that, then plant over it. The different cover crops have different types of roots for breaking up the soil. For instance, dandelion have long deep taproots; they help break up heavy soil and pull nutrients down to deeper rootzones. And something like clovers have shallow roots, to help with moisture retention closer to the surface. This is why companion planting works so well. Breaking up and feeding different root zones
Do you have interest in redworms or vermiculture? You could always add some redworms to the list of things to help aid in getting some great microbes and life in the soil and compost, plus snacks for the chooks