I recently measured the stock controls on a Vintage Modified P, and the values were very low- the tone control was 200K, and I think the volume was 220K. That really loads the pickup down.
The 500K Bourns pots I replaced them with were both on the high side, and it really woke up the top end. So did levelling the thick polyester finish on the neck- that glop is extremely thick. I didn't take it back to bare wood, just removed about half of the finish. It feels much more resonant and lively now, and it's evident in the tone.
Someone else mentioned pulling the tone cap- a better solution is to leave the cap in place, but lift the tone pot out of the circuit altogether- that gives a moderate upper mid and treble boost, helping articulation without changing the basic P-Bass tone. Back in college I put a mini-toggle on my '64 P-bass. It gave me the best of both worlds, because I could preset a darker tone for R&B, etc. with the tone control, then bypass it with the switch for more aggressive songs.
The biggest thing, is to ditch the flats. That is NOT the only tone available from a Precision. Think Entwhistle and Roger Glover when they were using Fenders, the list goes on and on...