Agree w/Myetball. I have the SONY factory system and few years ago replaced all my speakers with direct replacement Alpine Type-Rs (still have the factory tweeters in A-pillars) and also Dynomat-ed the crap out of the cabin. While the sound is admittedly not amazing, it is also in the order of magnitude better. I can now raise the volume to ~1/4 scale, the sound be much louder and still very clear. 1/4 may not sound like much but past that the rest of the set up shows its severe limitations and SQ falls off very rapidly (for reference I like EDM-type music, with a lot of low-end beat and my bass set to almost max). I like loudness but not at the expense of fidelity. At 1/4 it is pretty loud while still being crystal clear, pretty tight and w/zero cabin rattles (ok sometimes, but just sometimes the damn front door switches will make a peep). On top of this, a better amp could help although I fear driving the new amp with still speaker-level inputs will simply exacerbate the problem.
Having looked at those OEM
SONY-branded paper 10W speakers if you simply replace the amp and crank things up you'll probably rip the cones out of their frames in short order.
As for replacing only
the rear woofers, that is likely not to make any considerable difference. The crap door speakers still get the full-range signal (in stereo mode, maybe in surround mode the DSP
splits the frequencies but it also makes the rest of the sound absolutely horrible so that's pointless) and raising the low frequency in the EQ will effectively raise it for all speakers equally.
So before wasting time and money on putting in a new amp, I'd suggest replacing all the door/shelf speakers first. The OEMs are really bad. It'll be a bit more expensive (I paid ~$400 for all 6 required) but may actually give you the results you're after. Otherwise, yea, you'll simply make more noise.