The JJ MEC set in my Thumb BO4 sound just fine to me, sufficiently different to the sound of my Fender P and Lakland. If I hadn't liked the sound of the MEC's I wouldn't have bought it.
They can be a bit clean and dry (until you dig in) but I find them less dry in the Thumb than in my old ash bodied Corvette. IMHO they suit the bass well, can't think of a reason why I would change 'em.
You might also want to think about the amp? Through my TE I'm happy - but I couldn't get a sound I liked through the house Ashdown MAG300 the other week.
Regarding the Jazz thing - Outtoplayjazz might be able to help you there....he uses a Vette $$, don't know if he uses it for jazz though. Don't play jazz me'self, but strangely enough I think my DJ5 would cover this best out of my 3.