You have one really good thing going for you, Tom: it's a hickory. They don't have the shallow roots that make gardening under a tree much more difficult.
Other plants you could grow:
Epimediums
Hellebores
Japanese Forest grass (Hakonechloa macra)
Dwarf crested iris (Iris cristata)
spring woodland wildflowers like Cardamine(Dentaria) species, Claytonia virginica, Thalictrum species, Uvularia grandiflora
