By any chance do you live in the woods or suburbs? I live in the woods and have an exterminator treat the exterior of the house and inspect the interior for any activity. Just when I think I can end the contract new activity shows up in another location. Per my exterminator you have to find the nest(s), and since I live in the woods that may take forever...just as we think we found it another one turns up somewhere else. As you I am not happy about using chemicals and keep reiterating that fact, so they use it sparingly...I hope.
Good luck, hope it works.

Karen, I live in the suburbs, but there are woods on one side of my house. It seems these ants only visit me in the early spring, and then go away in the middle of summer. But you are right, I need to find the colonies to rid them altogether. If it gets any worse, I'm going to have to call the exterminator, to at least do the outside of the house. Thank you for your info. Karen

Been there, done that. Before I knew they were carpentar ants and would see them come out just before spring hit I thought nothing of it....until 1/2 of my friends house had to be torn down to the studs and have everything rebuilt!
I tried everything but the exterminator and it dodn't work.
Then I found a recipe that they loved to eat and I haven't seen one carpentar ant since.
it is:
Dark black strap molasses mix with yeast and boric acid in to a thick paste. Put this paste onto paperplate near the ant trail. The molasses attracts them, the yeast eplodes them and they take back the boric acid to the nest to kill off the others.
When I placed these on the kitchen counter it was like a feeding frenzy, I didn't count, but it looked like an army of ants.
Now, very spring I make this mix up and put around the kitchen and before I mulch around the house I get a container of boric acid and line all around the house outside as a barrier, this also worked for me in the old semi dirt, semi concrete hand dug this ole house basement.
Hope this works for you too.