I've done a full system upgrade but still have the stock 130 head unit. You will need to remove the stock sub because it is crap. It is flimsy and has a small magnet, no travel, etc. I replaced mine with a Kicker Comp vr 2ohm DVC. I put it in the stock box and had to make a spacer ring for it from 3/4" MDF. At first I ran it off the stock amp, which operates at 2ohms. There are two plugs because the stock sub has DVCs as well. It was barely noticeable.
So I did a ton of research aand ended up buying the Rockford Fosgate Punch 300x1 amp because of it's size and the fact that it is 1ohm stable. Much improved!! You'll need to run your power wire and ground like any other sub but this amp has speaker level inputs, aka hi-level inputs. There is a harness that comes with the amp that will allow you to splice the sub wires into the amp harness and this provides the signal and because it's speaker level it will also serve to power the amp on and off.
As for the two subs, they way you mention keeping the stock and adding one, I would advise against. The stock amp will also not add more to the front speakers. If I were to do it all over again I would have ordered 2 of the same Kicker subs but gotten 4ohm DVC so I could wire them in parallel to 1 ohm. This would give the right power to both subs. I would also get a box for them rather than using the stock enclosure. Don't get me wrong, it is 100x better than stock but there's more to be had. I was trying to stick with the stock HU but I'm going to replace it with a NAV unit with backup camera to help me see past the 35" spare and improve sound quality.
I know that's a long response but here's what I would do. Tap the sub wires into the RF Punch 300x1 and go with a 10" or 12" sub that has 2ohm DVCs and can handle 250-300 watts RMS at 1ohm. Hope that helps.
2012 JKU, 3.5" Rubicon Express lift, 35" Nitto Trail Grapplers, Pro Comp 17's, Smittybilt SRC rock rails, XRC front bumper with XRC8000 winch, Wurx rear bumper, and other stuff that's cool!!!