I rely a lot on evergreens of all types to help the garden for the 5 months of wintery weather. I have ground covers that look good until buried in snow, which here is often Nov.-Dec., low shrubs as Marie mentioned, and taller evergreens, both needled and broad-leafed.

In addition to Geranium macrorrhizum, there is a hybrid that stays evergreen, Geranium x cantabrigiense - I know of ‘Biokovo’ and 'St. Ola'. There may be other Ajugas as well.

Dianthus 'Greystone' stays a frosty blue-green for me all winter, so contrasts well with other plants and is fine in full sun. Bluestone carries it if you can't get it locally. It spreads slowly as stems root.

Carex 'Ice Dance' stays green and white in winter, but mine gets only 5 or so hours of sun and then shade, so I am not sure how well it would do in all day sun. There are other evergreen sedges with various foliage color and texture.

I have grown perennial candytuft in full sun and part shade, though it doesn't like it dry. Choose one with shorter, tidier foliage as some varieties can be sort of rambly. It grows relatively easily from seed. I am not sure if it spreads or each plant just gets larger, but my current ones were three and now take up a space that's a bit over a foot wide (limited by shrubs) and perhaps 5' long.

Another one that likes full sun if you can keep it in ordinary garden moisture, not dry, is common cranberry, Vaccinium macrocarpon. Mine turn deep red or mahogany in winter and keep the cranberries until critters eat them - the berries are crazy large compared to the fine-textured leaves. Stems root. Vaccinium vitis-idaea, mountain cranberry is a 3" tall shrub that slowly spreads.

If you are mild enough, some thymes will stay green. My wooly thyme stays partly green here.

Many types of Sedum and Sempervirens/hens and chicks stay green or gold or reddish in winter. I am particularly fond of Sedum 'Angelina' which is orange, chartreuse, and gold in winter.

I have a mat-forming Saxifrage with spring white flowers that has grown over many years from a 4" pot to fill a space that's 3' wide. No idea of the name since it wasn't labeled, but it is truly ground-hugging.

There are always Heucheras, Tiarellas, and their crosses Heucherellas. How decent their foliage looks in winter depends on situation and variety. Many won't do well in all day sun. Some types are clumping and some spreading.

Penstemon pinifolius, a SW plant, stays semi-evergreen here and may stay fully evergreen in your warmer climate. It likes it well-drained, and is available with bright red or yellow flowers. It has a delicate texture, and leaves stay around 6"-8" with taller flowers in late spring.

There are two native evergreens that spread, though only one really is a groundcover. Mitchella repens/partridge berry is a ground-hugging spreader. Gaultheria procumbens/wintergreen is looser-growing and about 3" high, but also spreads. Both have tiny white blooms and red berries. Wintergreen turns mahogany in winter while partridge berry stays green. Partidge berry prefers a fair amount of shade, while wintergreen grows in more than half day sun in some spots here.

Veronica 'Georgia Blue'

Some low-growing shrubs in addition to Marie's suggestions:

There are short rhododendrons, such as 'Purple Gem' and 'PJM Checkmate', both of which stay under 2 1/2' or so. Not spreading like a groundcover, but nice winter texture.

Check out some of the types of low cotoneaster which are evergreen such as C. microphyllus and C. dammeri. Even the ones that aren't evergreen may have persistent berries.

Bearberry/Arctostaphylos uva-ursi is another low, spreading shrub.

Heaths (Erica) and heathers (Calluna) are shrubby flowering evergreens that range from a few inches to a couple of feet in height. They mostly don't spread a lot, but planting a grouping of them would look nice all winter as a groundcover. Depending on the varieties you choose, you could have something blooming from February or March through October or November. They like full sun and sharp drainage once established.

Is rosemary hardy where you are?