If you want to build soil without investing much in a cover crop, consider annual ryegrass. This is from the website of the Sustainable Agriculture Research & Education. For the whole page, click here.

The summary goes on:

A quick-growing, non-spreading bunch grass, annual ryegrass is a reliable, versatile performer almost anywhere, assuming adequate moisture and fertility. It does a fine job of holding soil, taking up excess N and outcompeting weeds.

Ryegrass is an excellent choice for building soil structure in orchards, vineyards and other cropland to enhance water infiltration, water-holding capacity or irrigation efficiency. It can reduce soil splash on solanaceous crops and small fruit crops, decreasing disease and increasing forage quality. You also can overseed ryegrass readily into corn, soybeans and many high-value crops.