I'm a tried and true Jazz Bass guy, and mentioned it earlier in this thread, but still hold strong to an 80s era Peavey Foundation. They are a really nice spin on the J thing, sort of like how the original design G&L SB2 (with the single coils) is. The word has been out on these for the last 7 years or so, therefore they are not the laughable bargain they used to be. One could find these everywhere for $200 or less for years, now they sit closer to $350-500, often with original hard case. US made, great hardware, the best pickups (Super Ferrites) and CTS/Switchcraft electronics. The single coil Super Ferrites are sort of like a slightly tamed MFD, they can be aggressive, or dialed back to a more vintage J thing. No matter how much I like my J basses, I like the Foundys just as much, if not more. The US Peavey guys used to not like us promoting these, as it was our little secret, but the secret is out, so whatever... Get one if you can. This 83-89 era easily compares to an American Standard Fender, the early 90s ones are more like a MIM or nicer Squier.
This is my main one, from 1984, with it's twin 84 and a pearl white 87 parts drawer mutt with a Patriot neck.
It's also possible to get a cool Squier VM/CV to punch above it's weight class. I got this CV 70s J5 almost a year and a half ago. Got a screaming deal on the bass from Sam Ash, they said it was used, but it was new, sealed in the box for $300. Really worked on it to get the nut cut just right, the neck angle perfect and set it up properly. It took about 3 months to settle in to the dry climate here, once that happened, it was really great it easily played as well as my US Dlx Fender 5s. No issues with the stock pickups, either. It was sooo good that it deserved some more tweaks...
I then installed a B-Stock EMG J5 set ($100) with a BTS I had in the parts drawer (which cost about $60 years and years ago). Also got it a black crinkle control plate ($15) that matched the pickup tops as well as proper, set-screw J Bass knobs as found on the US models ($12 a set). Took two sets of knobs to have that extra small knob, so I have two large size spares. I also drilled and moved the EMG supplied jack to the side with a football plate ($5). So, it was $200 more in parts. Well worth it! Not all the modern Squiers are as consistently good as the ones built 10-13 years ago, but this one turned out great, guess I got lucky!
I have also played a few of the same model, but in a 4 string. Very cool basses!