True vintage tone, as in 1960's Jazz basses, came from pure single coils, which, as far as I know, were the only Jazz bass pickups of the time. Vintage-voiced single coils are typically bright, with more inherent treble than hum-canceling variants. The richness and warmth came from the amps of the day, flatwound strings, and rolling off the treble. Listen to Bob Marley and early Led Zeppelin. Those are single coils. If you roll off the treble on your MIM Jazz, you should get something more like the "vintage" tone in your head. (Overwound single coils, like Basslines SJB-2 Hot for J-bass, will have more lows and low-mids, and less treble. But those are not vintage-voiced pickups.)
The same is true of P-basses, by the way. Vintage-voiced split-coils have lots of inherent treble and clarity, but the stereotypical "vintage tone" comes from using flats and/or rolling off the treble.
Edit: For a good explanation of this, check out DiMarzio's description of its Area J pickups, which are designed to produce vintage tone with no single coil hum. Look at the "sound" chart, and notice how they actually have lots of treble and relatively less bass: [Invalid or Expired Link Removed]