Short answer: No, not at the same volume you are experiencing the rattle.
So the deck is powering the dash and the rear speakers. The dash speakers are crossovered at a higher frequency than the other speakers. The rear speakers are full range frequency. The door speakers are mid-bass drivers ran off a small amp in the trunk. The door speakers are crossovered at a low frequency. The stock deck has RCA preoutputs. (wow!) So adding a sub is easy, but won't affect the sound coming out of the other speakers. You would have to add a new amp, and/or new deck, or signal processor, or add crossovers to each speaker.
I am installing 6 new speakers, two crossovers, an amp, and a powered sub this weekend in my FR-S. I have been doing research into what it will take to "fix" the sound. I hope this helps. Good luck.
