Have any thoughts about how you'd want the tone to change?
I'd actually think about trying a pickup change in the existing body first. If your bass is like what you get when googling "Hagstrom HB-8", you might find something that fits in the existing routing or extend to pickguard/mounting variations - P-coils in a soapbar, etc. If you want something more trebly/snarly, a different pickup can well be the solution. When you're experimenting, you might bypass the pots and go straight to the output to see how that sounds - the existing pots might not be a match to a different pickup.
I don't think that a bolt-on configuration necessarily is negative on tone. A great instrument has the right match of neck wood properties and body wood properties. The problem is that you can't quantify the neck in any useful manner, so it's really a crapshoot finding a happy match in the body. I believe that this is kind of a physical impedance thing - the different densities and stiffness of the body and neck woods cause different energy transfer when they join at the neck joint. A good match allows everything to vibrate in harmony; a "bad" match causes an imbalance of the combination of body and neck. Maybe that means less sustain or a thumpier attack - of course, that might be "good" for some players!
On a through-body like an Alembic, the physical impedance match is always perfect because both ends of the string are on the same piece of wood. That means more sustain and preserved harmonics - which might be "bad" for some players!
The obvious alternative body wood would be alder for a more Fender-styled tone.
David Fung