Connecting new GFCI in the middle of a circuit run
Any branch of the circuit whose hot and neutral are plugged into the &quot;Load&quot; terminals, will be protected from that GFCI.
You are required to label those outlets &quot;GFCI Protected&quot; (NEC 110.3(B) and 8(C) of instructions).  Use any labeling method that isn't handwritten.  Also mark &quot;No Equipment Ground&quot; if applicable (NEC 406.4(D)2).
You can attach any number of branches to the &quot;Load&quot; terminals.  If you exceed the number of provided lugs, you can simply pigtail.
&quot;Push-in&quot; aka &quot;Back-stabs&quot; are generally not seen on UL-listed GFCI devices; I suspect that's a UL requirement.  Instead they have &quot;screw-and-clamp&quot; or &quot;insert wires; tighten screw to clamp&quot; terminals.  A bit tricky and you have to clamp HARD, but very nice when you get the hang of it.
If you have a GFCI that does have back-stabs, search it very closely for a UL, CSA or ETL mark with 6-8 digit file number.  If you don't see that, it is a knockoff/counterfeit device (which would defeat the purpose of installing a safety device, obviously). Counterfeits are generally not seen in competent retail shops; 99% of them come from mail-order, and the #1 offender is the Amazon Marketplace, which is eBay-tier junk mostly.