Wooly thyme. There are also other creeping thymes you can use -- perhaps do sections in different varieties for a little visual variety.

Kinnikinnick might do well in your climate. I've found it's a bit fussy at first and takes a while to fill in.

How about coral bells/ heuceras? I've planted them in rock garden situations and if you plant them close the will cover the ground and they remain ever green.

Wintergreen is a very nice evergreen groundcover. It spreads slowly. It has glossy leaves like Kinnickinnick and has flowers and red berries.

Aggressive spreading is the hallmark of a groundcover - it is this characteristic that makes them suitable for this purpose. "Invasive" aggressiveness can sometimes be limited by growing the GC in less than ideal conditions. For example, vinca grown in more sun in my woodland garden is not the slightest bit invasive nor does it smother other plants. OTOH, English ivy is abit too well adapted to too many growing conditions and behaves very aggressively in all but the most extreme situations.

I sell over 40 different species of groundcovers, the majority of which are evergreen. Not all will be hardy in your climate zone and not all will grow well in all situations. Some are better suited to pathways and between pavers, others best used as an underplanting for larger plants. Not all of the above suggestions - while all evergreen - will work well in all situations. Indicating what type of plants you hope to combine it with and the light conditions it will be growing in will result in some more specific choices most suited to your specific situation.

Starting this winter I decided to try 6 different evergreen groundcovers doing wintersowing. The 6 I tried were Wintergreen, Roman Chamomille, Dianthus Deltoids, Creeping Thyme, Bergenia Redstart and Mazus Reptans. The Bergenia, Mazus and Wintergreen are just now getting some sprouts and not that many. I have already planted the Thyme and Dianthus (as a matter of fact the Dianthus is already blooming). And the Chamomille I planted a while ago and today I planted a second go around on them. So far the Chamomille is looking the best. I will let everyone know later on how they all survived the hot summer.

JP