It says in my handyman book that I finally dragged out after all this fuss, that what I want to do is called "middle of the run" as opposed to "end of the run". Shows the three wires in a wire nut. I see the better connecters posted above, will check that out. Thanks.
So the word I wanted was "run", not "string".
I am not convinced there is no series vs parallel with all this. ISTR something about that in another handyman book, but have to check up on that. EG, if at the end of run you ran two more, but both from the same previous receptacle's second set of connectors.
Whatever, I have enough info to proceed and not electrocute myself or anybody else. Thanks again.