The F. Jazz 5 string came out in the 90’s, Ibanez Sdgr seemed popular , but nothing much stands out from then, perhaps I didn’t find the music that inspired.
Oops misread that you’re talking 00s not 90s sorry.
I use a basic pair of Sony headphones plugged into a Blackstar Bass Headphone amp. This plugs straight into the bass’s Jack outputs so no need to use your practice amp. It also has a fairly crude range of rhythm tracks and an auxiliary so you can plug MP3/ phone or whatever your source of music is. Cheap is h and effective solution.
After 20 years playing a Jazz, I had a break from all Fenders, but now gone back to a Precision amongst others and much prefer it. It’s lighter and I think sounds better than the Jazz.