in effect this should make it even clearer, as the sound going to you front speakers from the head unit will be seperated by the crossover fitted, and then the sound sent to tweeters/door speakers, however, as you must have noticed as you are changing it, the sound system is a budget one at best, therefore meaning that the crossover is probably on par with the rest of the standard equipment, therefore you have two sets of tweeters which will be giving out clear sound.
you could get a system wiring diagram and find the original crossovers, and then rewire the original tweeters to the new crossover box, or disconnect them completely and see if that makes it better?
try messing around with the head unit, as many have a lot of setting about moving bass around, also, with still standard rear speakers this could be a problem why you can't tell.
you could get a sub and amp in your boot, or there are 'subwoofer' 6 x 9's that are available, think they are about £200 from halfords, so search them on ebay to get them cheaper, supposed to be very high quality.