If you're using a Jazz, try TI Jazz Flats. I'd advise that yopu leave the tone wide on the bass wide open, shut off the bridge pickup and compensate for bass with your amp setting. Play near the front pickup or even at the end of the fretboard. Thumb works for songs that don't require very fast stuff, but two fingers gives you almost the same tone and better ability to play fast.
Before you go and spend money on new strings or a new bass, try playing the notes in a different position. Play on the lower strings, higher on the neck. This will give you a tone that's a little closer to a short scale bass sound.
And of course, try the other techniques already mentioned like using the neck pickup and plucking closer to the neck rather than the bridge. Both pickups together on a Jazz might work well, too, because the parallel wiring tends to reduce the midrange a bit. And finally set the tone knob to taste.
For the past few years I've been trying to approximate a "warm" thuddy short scale sound. I've been playing in a Rolling Stones "tribute" band.
For the past 10+ years I've played "Fender" style basses (a Jazz and a G&L). I've also used a Thunderbird and an EB-0.
Trying to approximate the SS sound on the "Fender-y" instruments is pretty difficult. I'm not willing to change strings, as I view that SS sound to be a more "niche" sound that I'm not using a whole lot when not doing the Stones thing- however, I will take a wild swing and say that I believe, that even with a string change, the best you're going to do is come somewhere close to the zip code of the short scale sound.
What I have found is handy for that tone on any bass is a foam mute. Although I haven't seen pix of McCartney using a foam mute prior to 1968- you do see pix of him using a foam mute on his #1 Hofner- as in the "Revolution" film clip as well as the early Get Back sessions. My theory is that he started using the mute on his Rick and liked the sounds he got out of it and transferred that over when he started using Hofner #1 again- until it was stolen.
As far as using my Jazz- I've found it does work best to use both pickups- with the bridge slightly rolled off and the tone rolled off and picking/plucking up by the neck join. I've done a lot of playing around with mutes- and the best material I've found for it is an old neoprene rubber wristpad cut down to size.
I took these pix when I was playing around with different heights- so what you see is a couple of different slices of foam stacked on top of each other. Since I took these pix, I've since cut a different section of foam- thinner and less obtrusive looking.
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In any case- the Jazz and the G&L really don't lend themselves to that tone- they'll work- just not as well.
OTOH, The Thunderbird and the EB-0 are good at it. The EB-0 is "best" as it is actually a short scale bass. Duh. The Thunderbird, being constructed mainly of mahogany, does have less natural ring and sustain. Again, with that- I use both pickups full on, and the tone backed off a little.
I've been playing around a little more with the EB-0, trying to figure out how well it works, vs. how I work on it. It's a really comfortable bass, but it's a different feel. For the most part- I use the Thunderbird for most shows.