We are really enjoying buckwheat. It is a super fast crop. Into flower in 30 days. We use it as green manure and also just to keep the weeds down, in areas we aren't planting in. The bees love it, and the seeds are easy to harvest.
I'd totally go with buckwheat too. It is a warm weather crop and fast growing, if not maybe the fastest growing cover crop. It will suppress weeds and then you can mow it down and till it under in time for a fall garden. For this purpose, don't let it go to seed. Let the bees enjoy the flowers, then mow it down after about 40 days.
If you are having problems with root nematodes annual marigold tilled in just as it flowers is a good way to rid your ground of them. Annual (not perennial) rye makes a good Winter crop to be tilled under as early in Spring as you can work the ground.
i'm a big fan of crimson red clover, the main reason is to hold back erosion during the winter months.
left was planted in mid oct, the right was planted in sept.
since our main garden is small and able to turn by april, we are never able to really benefit from introducing of the full potential of it. but it is one of the highest nitrogen fixing cover crops out there..
I'm also a red clover fan but for different reasons. It is a high nutrient forage crop for my rabbits and the extra, if there is any, can be dried as hay that is every bit as good as alfalfa.