While Field, or Winter, or Cereal Rye does germinate quite well in cooler weather I' be surprised if even that would germinate in December. I seeded some in early November here and the weather during November was too cold for good germination even then. Oats might germinate in cooler weather but temperatures below 32 degrees will kill and growth which would defeat your purpose.

You might be further ahead, to add a sandy compost and till it all together. Keep adding the sandy compost until the texture is good then plant annual ryegrass for the summer.
As fall approaches till it up again and put in a nice lawn.

I strongly disagree with any suggestion that involves tilling a lawn. After reading these forums for years and years, tilling is the one thing that stands out as being the biggest mistake found over and over again in magazines and books. Besides killing the long strand beneficial fungi in the soil, tilling fluffs up the soil to an uneven depth which eventually settles into a bumpy lawn.

What Mother Nature does in the fall/winter is drop leaves as a mulch. Mulch provides a blanket to keep the soil moisture level and temperature relatively moderated (compared to bare soil). This is ideal for fungal growth, which is what you need more than anything to get healthy soil. It is never too late to apply compost at a rate of 1 cubic yard per 1,000 square feet. If you do that now and make sure the lawn gets at least one inch of water per month until the summer heat hits, that should go a long way toward repairing your soil. You might also apply alfalfa pellets (any time) at a rate of 100-20 pounds per 1,000 square feet. Alfalfa feeds the soil microbes which are critical to making better soil.

Have you had your soil tested to know how much lime to use?

Also it would help if you added your location to your gardenweb profile so we can know where you live. Zone 6 stretches all across the country but the soil and climate conditions vary considerably from place to place.