If you're looking for something capable of some super-aggressive tones, but can still deliver a more mellow sound, try out the Blacktop Jazz. Here's the post I made when I got mine earlier this year.
But to put it briefly, soloing the neck pickup gives you a solid, if not an exact, Precision sound. When you start bringing up the bridge pickup you get a deep, cannon-like "Jazz on steroids" sound.
Budget-wise, they go for $500 at most retailers. Personally, I'd avoid anyone trying to charge more.
I've thought about getting a Modern Player Jazz, specifically the first generation (V/V/T config). I've tried them out at my local Guitar Center, and it's got a pretty tough-yet-smooth sound, but the Blacktop is still ballsier.
EDIT: I can't pass up the chance to also sing the praises of my Crafted In Japan (some are stamped 'Made In Japan') Fender Aerodyne Jazz. It's quite a "best of both worlds", what with the P/J pickup configuration. It's very light, if that's a consideration, sexy as Hell to look at and fits in with your budget. New ones go for $800, but there are a few used basses here-and-there that fall within the $500 to $600 range.