The thymes I have grown often take time to establish, but once the do, they form their own enduring little communities. I've never seen them become invasive here in the N. American Southwest, but they do establish with some extra care for that first year or three.

On a limited budget, I go thyme hunting. Next time you go to those public gardens that have the thyme you like, carry a sandwich bag with a piece of wet paper towel or rag. Tug a plug of the coveted thyme small enough that it doesn't damage the aesthetics, but big enough to work with, taking care to get as many roots as possible. Wrap those roots and the stems they are attached to in the paper towel and stick your Irish-gotten gain in the plastic bag. It is now safe to carry in your pocket.

I call this method, "Irish Gotten" because within my Irish ancestry, no human has the right to bogart Mother Nature's bounty. It is offensive to even ask a human if you can take a clipping or a plug, though a prayer of thanks to Mother Nature is always appropriate,

For what it's worth:
Very few of the ground cover thymes are great for cooking. The culinary thymes tend to be low, moderately woody herbs. The same Irish-gotten techniques will work for those thymes, as well, but carry something to snip the new sprawl branch from the main branch it is growing from. Irish-shopping protocol requires a benevolent harvest which leaves the mother plant healthier for the taking.

Thanks that's a good idea. I had no idea how to get it home ok without using a bucket and shovel. I think someone would notice that lol. I've dug up stuff for free in the woods or along roadsides but private property is private property. I don't like when people tresspass on me and damage stuff. I may see if I can find anyone who has some established thyme and get some.
Then I suppose I should try to eat it first. I'm leery I may be allergic to certain strains. I'm highly allergic to rosemary, but no other herbs so far. Which is strange since many herbs are related.
Anyway thanks for advice.