Assuming this is a passive bass, maybe you could simply try to replace the (presumably) A250k tone pot with A500k? That would "open up" the tone a bit with the tone control wide open. I have a feeling this might be what you're looking for. The downside is that the lower range of the tone control gets squeezed into a much shorter physical travel of the pot, which can make it more fiddly to precisely dial in darker/warmer tones.
You can replace the volume pot too of course, for an even brighter tone, but personally I find 500k volume pots too bright/spiky/harsh sounding with P-basses, and it's a kind of brightness that can be difficult to dial out without ruining the tone completely. If you like a bright/zingy/edgy tone it might be worth a try though.
Or you could take it much further and replace the stock pickup with a more "hi-fi" sounding one, with a wider frequencey range, but then it won't really sound like a classic P-bass anymore. When only replacing the tone pot you will still have access to all the tones you had access to before, just cramped a bit tighter on the pot.
I guess it depends on whether you want to retain a classic/vintage P-bass tone or mod the bass into something completely different. And of course, simply changing to a completely different type of strings will alter the tone FAR more than changing to another P-style split coil that's sitting in the exact same position.