Lots of options but if I were to categorize:
Faithful Reproductions/Copies
These are other companies builds that look and sound essentially identical to Fender Jazz basses, except the headstocks and perhaps hardware and electronics choices. Some diverse finish options. These are close to being Fender Jazz basses built by somebody else. They may or may not be parroting design elements from certain time periods in Fender's history. Examples are the G&L JB, the Lakland 44-60, and the Sire basses on the lower end.
Modern Renditions
These are definitely Jazz-inspired designs, but are "updated" according to the company's vision. Many are slightly downsized, as the traditional Fender J body is rather large. Many have active electronics, which has become so popular that high-end Fenders have active electronics now as well. Sadowsky is probably the most obvious example (though they will make you a full-sized bass too), and the Lakland Darryl Jones is a more significantly downsized passive design.
"Jazz Basses" with varying design cues
I put "Jazz Basses" in quotes here because some may argue if they fit into the genre. These are basses with two single coil pickups in roughly the jazz bass locations, but built by companies who wanted the bass to look like their design and not necessarily like a Fender copy. Pedulla Rapture, possibly the Fodera Emperor. Very arguable category, but the two single coil pickups will sound something like a jazz bass. Sometimes it's an alternative pickup option versus a dual humbucker setup that the company offers if you want one of their basses but would rather it sound something like a Jazz bass. Zon Sonus for example.
And a zillion boutique makers who align roughly with one of those categories.