My first bass was an Ibanez and if I recall it had jazz pickups and was in that price range, they might be a good brand to check out for that range.
Or actually, if you're in that price range, you might be able to get a fender or musicman if you are willing to look for a used one. Try and find one in the store and in person so you can test it out if possible, but you can potentially save between 1/3 and 1/2 of the original price of a very good guitar by picking a used one, and there should be used models where the worst damage the previous user left would just be a couple of pick nicks and scratches here and there. I got my current warwick corvette used for about 800, where it would have been around 1200 new, and other than a couple of scratches, it's in perfect shape--and I saved 400 dollars because of that simple fact that it was used.