Random Hot Leads in GFI Box

So I've been trying to trace why a kitchen outlet and outdoor outlet (in close proximity to each other) no longer work since my kitchen remodel last year. About 3 feet above the no-longer-working outlet is a new GFI that works fine. Suspicious, I pulled off the plate and unscrewed the GFI outlet and found a BUNCH of wires in there. Before investigating further, I shut power to the GFI at the breaker and ran my current tester over them to be sure they were cold. I think pulled out the outlet (which had 3 black/3 white pigtailed together and singles to the GFI. All well and good. Then I noticed that deep, deep in the back of the box was another set of leads (1 black/white/ground.) Low and behold, these were still hot. I suspect that when the kitchen was remodeled, the construction guys found these wires, didn't know what they powered, so capped them, stuck them in box and ran new romex for the GFI. And hoped that I didn't find out what didn't work anymore until they were gone. Honestly, I'm unsure what to do next. I'm hesitant to attach a hot lead to a GFI that already has it's own power and it's own breaker. Can I do that? Suggestions?