Series/Parallel mod. Trust me.
I agree, except that the OP said his bass was
already wired in series. (Both of my Jazz Basses have a series/parallel switch, and I go to series for more punch.)
This whole "which cuts through better-- J or P?" topic is endlessly fascinating to me, and I've lived it for years. It's not so simple-- the "Jaco sound" (Jazz Bass with bridge pickup soloed) and John Entwistle's P Bass sound (roundwound strings, tone control all the way up, percussive attack) are both aggressive, but the Jaco thing is "pointy" and the Entwistle thing has more "grind" over all frequencies. Will Lee feels that Jazz Basses are more audible in a mix, but I'm not sure one would have been in the Who.
The "neck pickup favored" Jazz sound can't really match the P sound, but I use it when I want a gentler version (in a duo with an acoustic guitarist, for example).
Strings obviously make a difference, too. The classic argument against the "P Bass is a one trick pony" indictment is to present clips of Jamerson and Entwistle, both using a P Bass, but the former with completely dead flatwound strings and the latter with fresh Rotosound roundwounds. WAY different. Then, you've got Aston "Family Man" Barrett playing thick, fat reggae with a Jazz Bass (flatwounds, neck pickup only) vs. Jaco with a Jazz Bass (bridge pickup only, fresh Rotosound roundwounds).
Then, as someone else said, you've got the context-- the other musicians' personal sounds.
Ahhh... it's an epic quest...