I suggest buckwheat as a cover crop and soil improver.Grows quickly and overpowers the weeds,and it makes a tremendous amount of biomass.An added bonus is the bees just go nuts over all the flowers.
I do have buckwheat down on my list as a summer cover crop, in case I need one, but have not purchased any seed.
I have rye, hairy vetch, and clover left from last year, so I’ll use that up first.