I'm not sure Alyssum (Lobularia) would stand up to foot traffic, but it grows fast and flops over, so it would cover ground quickly. It reseeds pretty eagerly in my yard. They are more like flowers than ground cover.
Often it looks like a snowdrift, say 5-10 inches deep if there is nothing to climb up on.
The variety that grew out of a seed mix for me either bloomed continuously or repeatedly all summer and into the fall (we have mild summers and mild falls).
Sweet Alyssum (Lobularia maritima)
