Thanks, all, for the responses. Obviously the best, and easiest way to improve the sound would be to chuck money at it, but that's currently not an option.
I ended up reconnecting the OEM tweeters, and the sound is now very good indeed. I had assumed that, as the aftermarket speakers I had bought were 3-way coaxial, they would cover the tweeter frequencies, but I guess there are some components acting as low/high pass filters, and so the new speakers were not outputting the tweeter frequencies.
In case anyone is interested the speakers I added were Pioneer TS-G1733i 17cm 3-way coaxial, for under £30. For that price I am extremely happy.