Well, that's a very interesting question. There are many possibilities for improving your sound, but most that I can think of would involve some expenditure of money, which sounds like you're trying to avoid. However, I kind of like your idea of using the 3.5 mm stereo splitter and using one side for your headphones and the other for some type of external antenna hookup.
Hey, I'm no expert, but it seems to me that you're not going to get into any trouble by trying your idea. First try your splitter into the radio, then plug your headphones into one side and see how that sounds. If it sounds ok, then try this, take a spare 3.5 mm stereo plug that's not attached to anything, just the plug with bare terminals, and stick that in the other side of the splitter and see how that sounds. If it still sounds ok, then try attaching your antenna (which could just be a long wire) to the bare terminals of the 3.5mm plug and see what happens. If it degrades the audio, then you can work on another idea.
Be sure and try various configurations when attaching the external antenna, such as touching only one wire to either of the terminals, touching both wires to the same terminal, etc.
Hopefully, somebody that actually knows something about his stuff will chime in with better advice, but to be honest, I think you've got a good chance of succeeding with your idea.
Good luck, and let us know how it works out.