Second attempt at posting this:
We grew a cover crop in the winter. To get full benefit with cover crops, you have to plow them under. My father described doing this growing up, but it may have started with soil conservation in the 20th Century. They used velvet beans, which he despised because it would drag on the plow. They also had small black southern pea used for cow fodder that they planted by the corn when they lay it by. The rows were 6 feet / 2m apart at that time, maybe due to using horses and mules to plow, and the peas vines would climb the dying stalks. After harvest, they turned in the cows to glean the field and they'd eat the peas.
He also told of a nearly forgotten cover crop: lupin. He liked it because unlike velvet beans, it quickly broke down in the soil, and it fixed nitrogen in incredible mounts. It was so common at one point that part of the US was called the lupin belt. Then came back to back hard winters that almost killed off the seed stock, and farmers switched to chemical nitrogen and never switched back.
We used rye after I was born. Rye puts an amazing amount of roots in the soil, which can help to amend it.