There is no hope for you. A head unit makes just as much difference as speakers do, hence the reason that Alpine radios sound better than Pioneer. Adding an amp will allmost double the amount of RMS power being sent to the speakers, making the music sound much fuller and richer. Making even Sony speakers sound OK.
If you add an amp to the Sony radio, the Sony radio is no longer responsible for the sound processing portion of audio reproduction. The only thing the Sony radio will be in charge of is the D/A converter, which has an effect on sound quality, and the preouts. Where the difference will be heard is switching to a radio with higher voltage preouts. In the $130 range, I can't think of any that go beyond the standard 2V preout. Now, the GT300 only has two pairs of preouts if my memory serves me right, so further expansion will be limited after adding a 4ch amp. That is, unless, you want to use RCA splitters. Buy a good Alpine 4ch, they run about $200, and you'll definatly notice a marked difference in quality of the sound.
Enough proof Burnrubber?? Shall I continue??