ORIGINAL: TheFingers
...Maybe what's missing is the pre-CBS Fender? Strings? Tape/Vinyl?
It's really hard to answer this question without knowing what you've already tried and a little more about what you mean by "woody."
I think Ed is kind of on the right track, instrument-wise. Have you ever played or seen someone play a bass that achieves the sound you're after? A lot of modern basses have a somewhat brighter, stringier, more aggressive sound than older basses, although that doesn't necessarily mean they're not "woody," as well.
I've been playing bass for about 20 years, and have played dozens of basses, and owned probably ten, and have recorded and done live sound for probably 100 different bass players, and to tell you the truth, I don't think that having "vintage" or boutique instruments or gear really makes that big a difference with most kinds of conventional bass. Certainly way less than it matters for guitar.
Technique makes a huge difference in the sound of the bass, and consistency and control matters a lot, much more so than with guitar, which is usually supposed to be an "expressive" performance. Good studio bass players adjust the way they hold the pick or finger the strings, the type of pick, the type of strings, the way they fret the notes, where on the string they play, and so on to control the dynamic profile and duration of each note.
For the sound I
think you're talking about, I would probably start with a split-coil pickup, maybe with a
tiny little bit of a jazz-style bridge pickup rolled in on the volume knob, if the bass has a second pickup, a well-broken-in felt pick played just behind the split-coil pickup, very slight palm-muting, the tone controls rolled about halfway down, and either groundwound or well-broken-in roundwound strings. Of course, technique is going to make the biggest difference.
I think the only thing gear-wise that should really hold you back is if you are playing a bass with a very modern sound-- hot, single-rail jazz pickups, for instance, will want to have a deep, slinky, aggressive kind of sound and probably won't cooperate.
Even a cheap bass can usually sound pretty good, and can usually get you pretty close to where you need to be. A better instrument or better signal chain can give you a better version of "that sound", to be sure, but a good player can usually get something in the ballpark even with subpar gear.
Messing with your effects or signal chain may give you a better version of the sound you already have, but is not likely to be satisfying if what you are looking for is really a different sound. Kind of like trying to get a fingerpicked stock telecaster to sound like Rage Against the Machine.
Cheers.