How to get a "woody" tone

Honestly, the woodiest tones I have ever gotten involved a regular bolt-on maple neck, alder body, passive electronics and a standard STAMPED old-school Fender style bridge.

High mass bridges, neck through bodies, active electronics, and super hard body or neck construction all serve to get further and further away from a woody tone in my experience. These things are not "wrong", but won't get you to the tone you're after.

 

In my experience the best way to keep an even tone as you move up and down the fretboard is with your right hand technique. You mention that you like your tone when you are playing frets 7-12. So my recommendation is to change your right hand plucking/position as you move up and down the neck, so your tone is consistent. In practice this means plucking closer to the bridge as you move up the neck, and farther from the bridge as you move back down the neck. Don't be afraid to move your right hand over the fretboard (for example try plucking with your right hand at the 12th fret while sounding an open string) if that helps you achieve a woodier sound. Does that make any sense?

 

Short scale?
If I understand correctly, what you like is the sound of shorter and more flexible strings (the fuller sound of the lower strings played in the second half of their range). So, to get that sound and feel, I would choose a short scale bass, preferably a hollow-body or semi-hollow, roundwound strings, and I'd play close to the fingerboard.

 

+1

Fender. Precision Bass. Maple Neck. The End.

 

I'd like to add...

Additionally, flat wounds help contribute a more woody tone along with what's mentioned above.