I am wondering if there is a flowering ground cover, or non-flowering one, that would come up after the tulips and grape hyacinths have spent their blooms. The area is full sun.

I have been working on this issue for a while now. I'll tell you what I've read, what I've done and what hasn't worked.
I'm anxious to hear from you, because I have some work to do in this area.
What I've read is to mix the bulbs, late winter, early spring, spring, and fall... you get the idea. Bulbs like crocus for early and lilies and buttercups for mid and then canna and gladiolus for late. What I've planted is just that.. . Plus some ground cover roses and perinnials like daisies and more statueque perinnials like lavender. Then when I feel up to it, fill in the bare spots with some annuals - or not. . I have gotten quite fond of thyme, other herbs, and sedum as a ground cover in the bulb areas, too. Chicks and Hens seem to be working in the drier areas of the yard. Ive been experimenting with wave petunias and geraniums, while they look great, neither of these winter over in my zone. I like a simple cedar bark mulch, too.
What I don't like - and this may just be me - is when the bulbs are too mixed up. Then it's hard to clear out the dead leaves after blooming is finished. So, I've redone my bulbs to 3 - 6 ft square sections of like bulbs that bloom around the same time. Then it's easier to clear out and prepare for the next round of blooming.
've been working on the ratio mix, too. Again, this is just me. I've been mixing the bulbs by size. For instance, If I have a 2 inch diameter lily bulb, then I use enough.. . Say dwarf irises or other bulbs to make up the same 2 inches of bulb size next to the lily and plant as directed. Since you plant at different depths, it seems to work out just fine. I've noticed that packing seems to be this ratio, but now I'm dividing them. So this is my rule of thumb.
One last thought, plumbego is a late blooming perinnial. I have early short bulbs ( grapes hyacinthus, crocus and dwarf irises) and mid tall bulbs (buttercups and gladiolus) mixed in for a full season of color.

No problem planting bulbs in with groundcover. Depending on where you live, when the foliage of the spring flowering bulbs dies back, the groundcover gets going and eventually covers the fading foliage.

I have periwinkle ground cover all over my garden (dark shiny green leaves and blue flowers) that start to bloom in the spring before the tulips and continue until late fall. My tulips, crocuses and grape hyacinthe come up through the periwinkle, which by the way is ultra easy to transplant anywhere in the garden.

Tick

IDouglas, where do you live?

Periwinkle is lovely. Here in the midwest, it only blooms in the spring, after the Crocus all the way until the late Tulips are finished.

It blooms a little again in the fall. Same schedule as Viola -- (Pansy and Johnny Jump Up).