With gunslinger on rex, can I attack more than just two different units in range if there are more than just two? (1st attack via attack action, 2nd attack via gunslinger, can I continue to gunslinger off of the additional attacks?)

That's not what the rules reminder text on the card or the keyword entry in the Rules Reference Guide says. Can you let us know what gave you the idea that it could possibly work that way?

I think the confusion is coming from the difference between an Attack Action (which triggers Gunslinger) and performing an attack (which does not trigger Gunslinger).