The best option, as you probably know, would be to grow a cover crop right after your main crop. The cover crop should have a mix of plants that produce nutrients and biomass. Generally, this means a mix of legumes and grains. I like to use fava, oats or wheat and an annual vetch. The cover crop should be left on for the winter, then dug in to the soil in the spring. This helps to prevent erosion during the season when you aren't planting. I live in an area with a very short growing season, so I make do with planting a cover crop over only 1/4 of the garden in any given year. When I turn the cover crop over in the spring, I don't plant spring veggies in that 1/4 of the garden. This way the cover crop can break down a bit before I start planting. One important thing to note is that you should not let the legumes bear fruit - I cut them back when they start flowering. When the beans develop, the nitrogen is being pulled from the roots into the legumes. You don't want that.
Now, if you don't think a cover crop will work for you, the suggestion of legumes is a good one. You won't get all of the nitrogen benefits if you are allowing them to set fruit, but you will get some biomass. French sorrel is another good one. It is a perennial vegetable that you can use first thing in the spring. After that, I cut mine back to the ground once or twice during the growing season, and let the leaves lay on the soil. You probably already do this, but other plant parts like onion and garlic tops and squash plants can be added back, as well. The only ones I'd avoid adding back to the soil would be brassicas, tomatoes and potatoes.