I'm sorry but neither my Precision or Jazz bass come close to this thing.
Well, I surely know what you mean. I've been through my fair share of decent basses, to be sure, and found some that got pretty close to that sound in my head.
Jazz basses, I think, came close. I've had sordid affairs with many a Jazz bass, and I still have one that I like a lot.
I'm so old, though, that my first Stingray was a 78 model, and it was almost new.
And you know what? That still wasn't quite it. It took me a long time to realize that the Sound In My Head was dependent on flatwound strings. So once I put flats on a Stingray, I had found what I didn't even know I was looking for.
Now, of course, I'm a bit of a Bongo fan, but that tone is still there, lurking inside the Bongo. Don't play a Bongo, by the way, unless you're prepared to spend whatever it takes to have one.
Friend of mine, bassist for the psychodots and the Bears, tours with Adrian Belew and that sort of famous thing? He's a dedicated Fender player and not a gear whore like some of us, but he foolishly picked up a Bongo last week at a shop and now he can't stop thinking about it.
Just a warning, heh heh.