Thanks everyone who chimed in. I also appreciate all the preamp recommendations.
Let me clarify some things, just for the sake of continuing the conversation (which I'm finding very useful and interesting). I already have the Colour Box, which I use as a DI box/EQ for the acoustic guitar gigs I do, so that's the main thing: since I already have it, this would be one less expense. My budget is very limited at the moment, so I'm really trying to make the most of what I have. In second place, as I said, I've been playing bass for a long time, but never active basses, which means - you all guessed it - that I don't really like the tone of active basses. That said, I think they have their place and also some of the gigs I've been looking at kind of ask for a modern, active bass tone.
All this is to say that being able to get in the ball park of such tone with what I already have would be ideal in my current situation. The only bass I currently own is a Precision (strung with flats just to make things even harder) and - surprise - I'm having a hard time making it sound modern, even with all processing I have within reach. So I'm thinking that swapping it for a jazz would be the best compromise at this point, as I'll be able to get close to the vintage-y P-bass tone I use most of the time and also get this more modern tone using the Colour Box.
I know full well that the Jazz won't give me the Precision tone, that there is no real best of both worlds in such a situation, and also that I'll probably have to ditch the flats, but it is what it is. If I were significantly attached to this Precision I currently own, I'd be more uncertain about going this route, but even though I always gravitate towards P-basses, this one specifically is nothing special and for a while I had already decided I'd sell it and get one that I bond with more when my finances improve. Also, I don't really see myself going full in on active basses. It's just that I want to get an acceptable tone for some gigs in a genre that isn't really my thing but that, at the end of the day, is paid work after all. The reason I'm saying all this is to ask one last question: perhaps if I simply swap the TI flats I currently have in my P-bass for a set of rounds and spend some time tinkering with the Colour Box, do you all think I can already get enough of a modern tone or is a jazz that much better suited for this (I'm not really interested in going, say, the Stingray route at the moment)?
In case anyone wants to get a better idea, here's an example of the tone I'm talking about: