With 18watts RMS, you're not even getting the most out of your Alpine Type-S. If you don't want to install an amplifier, just keep the Type-S or perhaps install Morel Maximos (supposedly they sound decent off head unit power). But all in all, the first step in getting better sound is to buy an amplifier. If you want a recommendation, one of boner amps right now is Precision Power Phantom P900.4.
Sonicelectronix offer them for $200 shipped. Honest 4x140watts RMS or 2x450watts when bridged. Enough to power pretty much any front speakers and a subwoofer with bridged second set of channels.
I would second the recommendation of the HAT Imagine speakers. I run them in bi-amp mode with a 2.2uF cap on the tweeter. However, it's a waste to run them without an amplifier IMO. Give them at least an honest 60watts, though the mids could take happily over 100watts RMS. They could be over your budget as the cost $250 new.