Thanks for all comments and replies.
As some people seem to miss stated facts:
- Yes, I have a good Jazz already. But it is an EMG-loaded, small body five string. It sounds good but not much like a passive four stringer. Both tone from wood/neck etc and the electronics is different and it is a different and somewhat harder instrument to play, technically for me as mainly a four string player.
- Yes, my Precision is very good but I see many songs in our band calling for the more refined tone of a Jazz.
I used to have a Squier Classic Vibe Jazz bass which I think was a decent bass for its price and certainly well built. The neck dimensions suited me well and the general tone was good. I also liked the pickups quite a lot - much better than the "bark" of the Geddy Lee and Yamaha RBX4 A2.
However, the CV Jazz had at least as much dead spot issues as any "real" Fender Jazz I have tried. Third fret A string C was weaker than normal and with shorter sustain, fifteenth fret C + corresponding tones on D and G were dead, too.
It took that CV bass about 6-8 seconds to drop 20 dB in output on the third fret C. (I do not have the exact numbers in front of me.) My Precision and current Jazz sustain twice as long with much better tone, too. Same brand (new) strings, same signal path etc.
(I do not need 16 seconds of sustain but I figured I need to set up some criteria if I want to compare things. The sustain "curve" seemed rather linear so that bass loses any given level twice as fast as my other Fenders. Actually my Danelectro Longhorn which is often said to have a "thumpy" tone had better sustain than that Squier. Same strings, by the way.)
On the CV, whole notes in C would die out and disappear very quickly when I gigged that bass (think "Jump", slow blues etc.)
I will not rush this search. If something nice comes along I will give it a try. G&L, Lakland, vintage MIJ Fender/Squier are on my list...