I spent the better half of my youth chasing a similar sound. Not those particular bands (they didn't exist then), but the sound you speak of was pretty common throughout the 90s in punk/hardcore/etc. It took me way longer than it should have to figure out that a Fender Precision is probably the key to the sound. A lot of that tone's body and presence seems to naturally fall in the midrange rather than just boosting the treble or going for a scooped sound (which can be close but doesn't have the same weight to it). Both bands you mentioned (and like, 90% of punk bands) seem to have one thing in common, which is a mere Precision Bass.
Alternately, a MusicMan or G&L L2000/ASAT has an even more growly take on that sound in spades.
While I don't know for sure this was recorded with it, the bass player was using a G&L ASAT during this time, and mostly played fingerstyle at that:
Although keep in mind I'm not saying you should toss the Jazz and run out and get a Precision instead. I ended up with a Rickenbacker 4001 at a young age and tried in vain to get that modern punk sound for a few years with various amps/cabs and it just never did it. In the process I fell in love with it for what it is, and 19 years later no amount of money in the world could buy me something I like more. I've literally grown up with it and have somewhat developed my own sound/tone rather than seek to emulate others.
That said, I bought a G&L L2000 Tribute a few years ago on a whim and it's downright hilarious how it just nails that sound without even trying to.