Maybe the passion flower or maypop (Passiflora incarnata). It is a climbing or trailing plant native to the eastern US that likes sun and well-drained soil. The plant has the most spectacular blooms, edible fruits, and the plant itself is used medicinally as a remedy for insomnia. (I included the USDA link on this plant.)

If looking for something more temporary (until thyme gets larger), you can try some vining veggies like squash, cucumbers, melons, or beans. Food plants are often herbs in their own rights and should be at home in an herb garden. For example, cucumber is known for its cooling action - got irritated, allergy-ridden eyes, cucumber slices on your eye lids feel absolutely marvelous!

Of course, there are climbing roses. Roses provide not only medicine for the heart and soul (who can be blue when roses are blooming!) but essential oils and vitamin C laden hips. Roses were in herb gardens for centuries.

FataMorgana

Here is a link that might be useful: USDA Passiflora incarnata

Not for the faint-hearted, but ivy will cover just about anything in no time flat. It's a herb, of course.

Jasmine (Jasminum grandiflorum) is pretty, smells lovely, and will cover anything in the blink of an eye.

You could also consider hops, or honeysuckle. What about something unusual like Codonopsis or Madder or Schisandra? Grape vine? Kiwifruit? Loofah? Pepper (if your climate is hot enough)?

For myself, I'd go for nasturtium for the brilliant backdrop of colour; or a briar rose, unless you want fruit or veges of course, in which case you can't go past the passionfruit. Or try a choko (chayote).

Mind you, for some of these, a wall 3ft high isn't very high at all! You could train a trailing rosemary up it, perhaps.

Daisy, my understanding is that the OP wants things to trail down, not up. S/he says 'trailing' and 'creeping down.' Have I got that right? In which case the prostrate rosemary will be perfect. Cross pollination is not an issue unless it self seeds from either plant. Prostrate thymes will hang down, as will nasturtiums. Prostrate juniper might do but could get very big. Lots of things will trail down a bit eg sage, oregano,marjoram, lavender, chamomile and othe cushion forming things. Courgettes (zucchini) would also hang over the front if you choose the right varieties. Strawberries? Another good trailer is Saponaria ocymoides.

Thank you all very, very much. Yes, Flora, you are correct, I need something that will trail down, because abutting the wall at the bottom is a concrete slab. (Eventually I will be jackhammering that concrete out, but I think that's a project for another summer!)

You have all given me some wonderful ideas to explore -- thank you!

Definitely go for the trailing rosemary. I saw some trainling downwards in Las Vegas that hung down several feet, but was also covered with bees visiting the pretty blue flowers on it. It was beautiful though.
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