My neighbor has a willow underplanted with soloman's seal and lily of the valley with a few jack in the pulpet here and there. They all do very well....and the solomon's seal adds interest even when it's not in bloom.
Linda C

We just moved into a new-to-us older home with a huge, stately willow in the yard that is underplanted with ajuga. It's beautiful, dark reddish-green, about the height of the surrounding lawn but never needs mowing.

Any drought tolerant shade plant will thrive beneath your willow.....and look quite nice at the same time.
Try to pick up the varigated look of its leaf when choosing ie-
hosta "Patriot"
pulmonaria "Majeste"
Gooseneck loosetrife
Laminium "Nancy"
Lamb's Ear
Feverfew
Viola(for spring colour)
Foxglove

to name a few

If you're also looking for shrubs to plant under it, Aucuba and Nandina are both reputed to compete successfully with tree roots. We don't have a Willow, but our Aucubas & Nandinas seem to resist competition fron anything else. Native Sword ferns seem to compete too, and Wild Daphne, (Dapne Laureola) certainly does, but I don't know where you'd get it. It's also called Spurge Laurel & I've read that it escaped into the wild when it was introduced here from England. There were several growing in our garden when we moved in about 20 years ago. They have nice dark green, glossy leaves & seem to grow slowly up to 3 or 4 feet high. They're tough customers - almost impossible to dig up without leaving some roots that then grow back. The flowers are inconspicuous and the berries are very poisonous.