Redo - cover crop or compost?

Without going into the history, I have decided to redo my lawn. I am in Central NJ, and I have concluded there are two ways to go, to be done this fall:
1) till up the existing greenery, smooth, level, weed, then till in 2 - 3 inches of compost soil, then seed with tall fescue, or
2) till up the existing greenery, seed a cover crop that will die in the winter and till it under in the spring, then seed with fescue next year.

There is the problem that northern grass should be seeded in the fall, so perhaps 2) would actually entail waiting until next spring to start, and killing the cover crop by mowing instead of relying on the cold.

Are there reasons to prefer one method over the other? Thanks.

the history is important to folks here
and a lot of us don't care much for the till it in and spray for weeds kinda stuff

so I'll start, whats wrong with what you have now?

I have never used chemicals of any sort on the lawn. But I also have not pursued a rigorous (or even mildly strenuous) organic care approach. Meanwhile the soil has been compromised by construction to our house - a few years ago parts of the yard were destroyed during construction, and the contractor repaired them with lousy soil. This was worsened when the inspector made him regrade the whole thing. I haven't tested the soil but the results speak for themselves. I also didn't know about mowing high, tho I did mulch. Then the incredible heat of this summer added insult to injury. So most of the grasses (contractor grade seeding) are now gone, replaced by weeds and crab grass. It's not a big lawn, less than 5,000 square feet, and I want to start buy doing just 1/3 or 2/3 of it this year.

Is the grass species you plan to use fairly shade-tolerant? If so, you might consider planting your cover crop as a "nurse" crop, mowing it back to 4" or higher (or cutting with a hedge trimmer) whenever it starts to flower, and composting the cuttings. The bulk of the benefit from the cover crop would be from the root mass dying back each time you cut, but you'd also get a lot of compost from that plan, perhaps enough to rake some of it back into your lawn.

If you chose your cover crop well, I bet you can get away with just using a pick to break up the most-compacted parts of the yard.

OK, that's interesting. I was planning to use tall fescues as recommended by Paul, especially as we may well have more summers like this one. If I understand correctly, in your scenario I would grow the cover all summer, mowing at 4" (I have a corded electric mower always set at the highest setting anyway), composting rather mulching the cuttings. Then have it die away over next winter. But wouldn't it then have to be tilled in before seeding? Also, would I have to remove the existing "greenery" before planting the cover?

>would I have to remove the existing "greenery" before planting the cover?

Not necessarily. Different methods, or even a succession of cover crops, can do different things at different seasons.

How does this sound:

Initial cover seeds are sown, and as they are just sprouting, the  yard is scalped. An initial, tall crop is cut and left as mulch. A canopy re-forms from a second growth of the cover crop, working with the mulch to shade any pre-existing vegetation almost completely. It might need cutting at that same height if it blooms again.

As that stage of cover crop loses vigor, and the original residue begins to rot away, nurse plants are established which will offer dappled shade, helping to give the desired turf grass an edge over any weeds that persist, but with the intent of allowing enough space that grass can become established. The yard is cut down to a few inches (and removed) when nurse plants are a couple inches tall, but growing fast.

Turf grass is seeded at the normal time of year, but maybe a little less densely than usual. Other sowing dates are timed around this event. Maybe the nurse crop goes in along with the grass seed, maybe slightly before or after, depending on how the two will interact. The nurse crop, or some species from it, might be allowed to re-grow once or twice, with a transition to frequent 4" mowing as the turf gains on the nurse plants. As mowing becomes frequent enough, the cuttings will be short enough that they can be left in place again.

Any plant that has survived all this, would likely also have thrived after tilling & compost application. If you notice any particularly stubborn species during this process, I imagine that hand methods will do more to stop it than if full sun, frequent irrigation & surface-applied fertilizer were available.

Details of climate & soil would help you to choose plant(s) for a scheme like this, and you might  have to compromise a lot from the ideal I've described, but if the short-term appearance of the yard isn't important, I think it's a goal worth shooting for.

So would I start this fall with a cover crop with no soil prep, let it die over the winter, then sow again in the spring, maintaining mowing to avoid the crop seeding, and tilling everything under in the fall, then putting in the turf seed? And do I till in the spring before putting in a new cover? I guess having never dealt with cover crops, I am little unsure of the methodology. On the other hand, I am cheap, so it is more appealing than a couple grand on compost....