Let me introduce you to another cousin of lavender, thyme, rosemary and sage: bugle (Ajuga reptans). I've seen it growing alongside other creeping groundcovers like strawberries (Fragaria) and mint (mentha), normally in rather moist soil under oak trees.
In the garden I have tried it scattered out through shady parts of the garden but I recently decided to transplant all the existing clumps together so to make a generous group. This way I've found they easily make a nice carpet of foliage and that all the tiny spikes of blue flowers have a much bigger impact. If you use it as an extensive groundcover it can actually tolerate moderate foot traffic and act as a lawn alternative for shady spots in the garden. It is also easy to use in containers in shade.