In my opinion a "modern" sounding bass is defined by a top-end glassiness and upper-mids kick that usually requires some level of active reinforcement. I think the Ken Smith/Cirrus/Yamaha TRB sound is another branch of "modern" that's more niche to certain bands or atmospheres, but doesn't necessarily cover the whole term. I would definitely think that a Sadowsky JJ or Fender Deluxe Jazz, with the bridge pickup favored, and some highs added, would be "modern" to these ears.
Also IMO, things that AREN'T modern, the absence of which thus supports a modern sound, are "woodiness" and a rounder, P-Bass like tone. When I hear tone that's obviously shooting for an upright-esque sound, I consider it "vintage," even if the performer and band are a "modern" band.
For a third time IMO, your best bet for getting "modern" would be to invest in an outboard preamp for the Jazz bass - maybe look for a Sadowsky outboard pre - and learn to favor the bridge pickup, both in the signal balancing and how you anchor your thumb.