I didn't keep them hooked up to the factory amp because Monsoon has two outputs per speaker - one with only low frequencies and one with only high frequencies. The factory speakers have four inputs for this, not the typical two. If you just choose one of those pairs of wires to connect your new speakers to, then you'll only get lows or highs, not both.
I bypassed the Monsoon amp completely, but this can only be done if you have an aftermarket radio. The factory radio has low-level speaker outputs knowing it will be going to an amp. Aftermarket radios have a small amp built-in. One thing I found out is if you have the GMOS-04 connected, it will make the speaker outputs low-level (also being designed to be hooked up to an amp). I had to bypass that for the rear speakers, so the actual radio amp would power the rear speakers.
As I mentioned, I'm replacing my front speakers as well, so I went ahead and got an amp that will drive all four sets of speakers. I will run RCAs from my head unit back to the new amp and leave the Monsoon amp connected to the front speaker inputs from the radio/GMOS-04 just to power the center channel (chimes will only come out of there now).
Here's my equipment setup. The speakers and amp are probably mid/low-mid grade, but I'm not an audiophile, so it will be good enough for me:
Head unit - Pioneer AVH-X3500BHS
Rear speakers - Pioneer TS-A6965R 6x9
Front speakers - PioneerTS-A1605C 6-1/2 components
Amp - Pioneer GM-6500F