sedum angelaina certainly spreads and loves full sun.

that's Sedum Angelina

I was going to suggest sedum as well. I have a few - John Creech, Westenphaner's (sp?) Gold, and kamschaticum. The kamschaticum seems the most hardy and vigorous of the three, all of which are in poor soil on a small slope in the hot afternoon sun directly next to the street.

Creeping phlox may do well here for you too, although IMO the foliage is not as nice as the sedum. But I don't have great luck with phlox, despite trying for over ten years. There is a Wendy's in town, however, which has a stunning display of it!

By the way, the sedum has a nice burgundy/maroonish color during the winter, which is an added bonus.

Dee

I have sandwort in my front garden. It gets full sun. I'm quite happy with it. It started as a 4" pot 2 years ago, and you can see in the photo how large it has become. This photo was taken about a month ago. When I just looked at it this morning, it is still in bloom - though not as profusely. And it's a bit bigger still.

Here is a link that might be useful: sandwort, arenia

My number one choice for evergreen groundcover (plus the bonus of Spring
bloom) is Vinca minor - in extreme drought, you might have to water it to
prevent scorching, but I've been using this for decades in full sun, even as a
grass substitute (No mowing!) between my driveway and the neighbor's driveway.

Other choices: Iberis, which is a sea of white blooms in Spring, but stays reliably evergreen the rest of the year; Phlox subulota; Ajuga; and as others have suggested, Sedum Angelina - just watch it closely if it adjoins any grass
area. . .you do NOT want it running in your lawn!

One somewhat accidental choice I've enjoyed: a mix of Geranium 'Wargrave's
Pink' and 'New Hampshire Purple' - a couple of plants just started filling in
at the base of some rose bushes and quickly became a bonafide ground cover.

Carl

i was going to say vinca too. I've heard many say it is kind of a slow spreader, but I think it does better in full sun. I put some in last year in terribly, sandy, compact soil right next to a hot, sunny driveway and it's done very well. It's filled in the spot I put it quite densely and seems to be spreading outward this year.

I've always had trouble with weeds in vinca grown in sun. Right now, I'm avoiding a patch that is full of the ultimate weed - poison ivy. Grass is much more usual, though, and that I'm willing to do personal battle with.

Given that this used to be a driveway, is soil compaction going to be an issue? I know I've run into lists of plants for heavy/compacted soil. You might want to grow some of those for a while, just to break things up, then put in the final planting.

These grow in full sun for me:
Cranberry as long as it's not dry, average moisture with organic soil is fine.
Veronica 'Georgia Blue' in my garden is in an average moisture area also, so I don't know if it likes it dry.
Mums - I have two, both of which spread. Chrysanthemum weyrichii (I have 'White Bomb', but there is also a pale pink type) is only a few inches tall until it blooms in autumn with 6" stems, and an unnamed variety that's about 1' tall with very double coppery colored flowers. (You are welcome to come get some of either if you are ever in central NH.)
Androsace - Many are fussy rock garden plants, but I have one that's been easy and has slowly spread in the 3 or so years I've had it. It's in a well-drained area. I'll try to find the species name for you. White flowers in spring, dense small leaves the rest of the year.
Sempervivums/hens and chicks spread slowly but densely enough to be weedblocking.
Thyme - wooly seems to work best for me as a groundcover, likes it average to dry.
If you want something taller:
Siberian Iris and the old-fashioned daylilies will spread, and with division will fill space relatively quicker. Average moisture or wetter for the Iris
Clematis 'Mrs Robert Byron' is a non-climbing clematis that will fill quite a bit of space on the ground a couple of feet high. Size-wise is often described as a monster.
Nepeta will root where stems touch the ground and so will fill space over time, more quickly if you move the new plantlets. Makes a dense, weed-blocking mound in my garden. (Again, if you are headed this way or want a garden field trip, I'd be happy to share.)
Lysymachia puntata/ yellow loosestrife
Old-fashioned German Iris makes a pretty dense groundcover in my garden, spreading slowly but densely over time.
I have another plant (I'll try to take a photo if it ever stops raining!) that spreads rampantly that I got from another gardener. Magenta double flowers with large toothed leaves. 1 1/2' tall. (I'll share - if you want it.)
Lambs ear/Stachys self-seeds to make a groundcover. Needs some care as far as cutting back flowering stems and will invade lawns, etc.
Mints, both the culinary mints and bee balm.

That's what I can think of so far. Some of these are sort of thugs, while others will play well. Depends if you want to fill space in a weed-blocking, soil-holding way or slower-spreading ground covers.

There are other threads on groundcovers over the last couple of years if you do a search.

well, if you are willing to take a risk, there is snow on the mountain, or bishops weed - although it's considered an invasive on a lot of state lists. But I think it's beautiful and always loved it and I've read differing opinions on invasive plants and the evolution of ecosystems...

but anyway, it is said to spread like crazy so it might thrive with the compacted soil.

There is also a variegated pachysandra that i always thought was pretty, and while I always thought of pachysandra as a shade plant, I see it in very sunny yards all over my town. In fact I do wonder now if my pachysandra at my old house never filled out well and always looked scraggly because it was in too much shade. I grew up in a neighborhood full of it so I like it, but I've recently heard of it described as "the vinyl siding of landscaping." I've heard some people find it invasive too and can't deal with how fast it spreads either.

Wow, Pixie Lou! I've never seen sedum used like Victorian bedding plants. That really illustrates the variety of colors available in groundcover sedums.

I have lots of sedum angelina...I love the bright chartreuse in the spring and summer and it turns a lovely rusty color in the fall and winter.

And it pops up in new places...

Another ground cover I have that does well both in shade and sun is my geranium 'st. Ola' - it does not die back in the winter like my 'rozanne' does and the leaves get a nice fall color. Plus it's spring bloom is amazing.

This was it in April - This area around the fountain is in full sun.

You might also consider filling the area with winter heath, Erica carnea. The foliage is sort of like miniature conifers, evergreen and feathery, but in February (sometimes earlier) the plants are covered with flowers in shades ranging from deep purple to pink to white.

They like lean, dry, acidic soil, so they'd probably be good in the area you're describing. They're especially wonderful in large drifts, too.

I LOVE these - mainly for the color they provide in the coldest months, but also because they need no care once they're planted. I mulched mine when I put them in, maybe watered a few times the first season, and have done nothing since then. For a few bucks per plant, they've given me about 20 years of enjoyment.

I really like the low growing sedums. There are so many different kinds in all different foliage shapes, foliage color, and flower color. I also love hens and chicks, so many different kinds. Both do really well in full sun.

I moved recently but I'll see if I can dig out pictures of the beds I had full of hens and chicks. They look really cool in the fall and winter months too.

You might want to consider making a pathway with some flat stones, or pavers, that these groundcovers can grow around. It looks really cool and it will also help you when you need to get in there. A garden ornament would also give it a focal point.