There is no easy way to improve your soil. To grow anything including a cover crop you'd have to sweeten the soil with lime and it would have to be dug before you'd consider a cover crop. In addition being on a slope you'd have to devise some way to hold the water or it would simply run off taking your soil with it. Have you considered terracing it and working on a section at a time

Thanks for the info. The slope isn't steep enough that I feel the need to terrace. Also it runs into the main area of the lawn, so the transition would be awkward. I think I'll just pile some mulched leaves up there in the fall, and hope that helps over the winter.

Have you tried growing things like lavenders, thyme, or thrifts in this area...

Not yet. Until this year I haven't tried growing anything other than grass there. This year the willow shrubs I planted are doing okay, but the area in general is quite bare.