A good option is one of the basses from the Squier Classic Vibe series. Every CV instrument I've seen is pretty high quality, and they have better pickups than the cheaper Squiers. If you get yourself a 50s Classic Vibe P-bass (has the single coil, not split pickup -- Sting is famous for using that kind of pickup) you'll also give yourself a slightly different tone to use when you want it.
I was in a similar boat a few months ago. I ended up searching Ebay for a "parts bass". I found a Fiesta Red P-bass made from Squier parts that had a Fender logo put on the headstock, and the guy upgraded the electronics. The pickup on it is from a Mexican P-bass. It sounds amazing for the 180.00 I paid for it, and I am not that worried about losing it at that price.