I have the Tele type 54 P bass with single coil pickup and I agree the stock pickup isn't too bad. Better than a lot of the really cheap junk out there. However I swapped it for a custom wound pickup from Pete Biltoft. He made it to my specifications and it's exactly what I hoped it would sound like.
I considered the Duncan Quarter Pounder, but that's the one I have in my 78 P and though it's one of my favourites I just wanted to try something different. The QP would be a very good choice and incidentally, I don't think it's at all scooped in the mids as some others have stated. I have to scoop a bit of mids out with mine. I don't know where that idea comes from. Even Duncan on their website states the QP has strong mids.
I just got an old G&L SB1 and the pickup in that would be a good choice for your Squier too. I sure like mine. It's a very hot pickup but I can mellow it out by backing off the tone control a bit. I use Thomastic flats on all my basses.
Experimenting with pickup height can make some dramatic changes so try that with the stock pickup first, You might not want to change.
I'm not particularly fone of DiMarzios. Had one in my 78 P before the QP and the QP was a big improvement. to my ears.
In the end though, it's your ears alone that you have to trust. You can't make a sound decision based on the opinion of others, so get out and try some P basses and find out what pickups you like.