When I owned a Jazz bass, my favourite ever sound was when I fitted some old (ish) DiMarzio pickups. They were just brilliant. Definitely more "oomph" than any other passive Jazz pups- but they had a lovely warm sound, which seemed to produce the whole range really well.
Plenty of low end, punch at the top end, and mids that cut through well. Whenever I played that bass, and in several bands / genres, it just sat nicely in the mix. Wish I hadn't sold that bass...
Re pickup height - the luthiers I've spoken to, and the ones I watch regularly on YouTube all say the same thing - don't set the pickup height too close to the strings.... so adjust that carefully.
Let us know how you get on 😀