The list is pretty good really, that 3-4 hundred I'd be looking at Classic Vibe Squier, Yam Broad Bass, Marcus Miller if you like a slimmer width at the nut. G&L were suggested and theyre right in that price range, I'd probs lean towards one of those.
Classic Vibes are great and you'll easily find one that suits your style. I will say, I just recieved an affinity today and I'm surprised how cheap it feels. I'd never played one before and the neck feels like raw wood, theres flex in it, Its a PJ and the Jazz pickup might as well not be there, its so weak. I bought it as a modding platform, but I wouldnt recommend it, the CV is light years ahead. I'm contemplating sending it back but as its a project bass, I'll be changing the neck, tuners, bridge, nut right away, and might swap out pots and pickup down the line a bit.
Used, i have a perfectly good Yamaha RBX270 that was under a hundred quid on ebay, and 400 should find you a decent mexican Fender in the classifieds. I had a little look at one that went unsold at 400 on ebay last few days. Theres one for sale near me at 450 that looks good. I paid 400 for my MIM P-bass and that came with a hard case and a practice amp. Did need a full setup though.