Hmm... I'm thinking clover. Sow thickly with cover crop seeds. There are other cover crop options but clover will stay low. Look at different varieties -- there are others besides the standard white clover, and even white clover come in different strains that grow taller or shorter, levels of winter hardiness, etc.
I opted for sweet clover with yellow flowers. The seeds themselves smelled sweet, and to my pleasant surprise, the foliage also smells sweet underfoot.
Clover will add nitrogen to your soil too. Make sure to get the innoculant for it.
Some people don't like the bees clover attracts though. With the approaching summer, another good cover crop is buckwheat. It will grow taller but will winter kill and add humus to your garden soil.
Oh! Another option is a Sunflower House. Here's a thread: https://www.helpfulgardener.com/forum/viewtopic.php?t=8140