Here is my dilemma. I have a fender American special jazz bass purchased in 2011. I used it heavily until 2018 and then started exploring other basses. Ended up finding my sound with Ernie ball music man basses. I have two 5 strings. Stingray special 5 and bongo 5. Both HH models. They became my two main basses. Two years ago, I decided it was time to change the strings on my fender jazz. I purchased a set of Ernie ball roundwounds custom gage 50,70,85, and 105 were the numbers on the strings. I thought I was using what came stock on stingray special basses. It was the first time I had purchased Ernie ball strings for a bass. To me my jazz sounds dead now. There is no where near the sustain that there is with my stingray and bongo and I don't remember the jazz bass sounding that lifeless even though prior to the string change it sounded dull compared to the stingray and bongo. What do you guys think? Could it in fact be the strings or could it be that there just simply won't be anyway to get a tone like the music man basses? I did try another fender bass and was also disappointed in its tone as well, and that was a Tony Franklin signature bass. There were flat wound strings on it though. I know this thread could easily fit on the basses thread as well.
That Jazz Bass Special, if I recall correctly, has a P-bass split pickup in the neck position and a Jazz bass pickup in the bridge position and it's all mounted on a P-bass body...correct? I would be surprised if you could get it to sound like a normal Jazz bass because of the P-bass pickup in the neck position. That's always going to give it a darker sound than a normal Jazz Bass pickup with fewer of the higher harmonics. And, unless you're Jaco, soloing that bridge pickup sounds pretty thin, but great for harmonics. So within those limitations, you're adding round wounds. What strings did you have on it before?
I use the LaBella White Tape Wounds on my passive 1982 Jazz Bass. For the 35-years before that, I used round wounds. First was the Dean Markley Blue Steel set that came on it when I bought it in 1985. I replaced those twice, so I ran three sets of those. Then I replaced them with DR Black Beauty strings and ran those until about 5-years ago, when they turned into DR mottled gray no longer beauties string. With both of those round wound sets, the D and G strings sounded pretty thing below the 5th fret. Above the 5th fret they sounded good. The DR strings sounded better than the Blue Steel strings, but not by a lot. Then, about 5-years ago I bought the Tape Wounds because of a video I saw on You Tube comparing a bunch of strings.
The tape wound were much better sounding to me than the flat wound strings, but not as bright as the round wounds. They also had by far the best balance of sound string to string. They do not have a thin sound on the D and G strings within the first 5-frets like most I've used, and I'll play an E on the D string if it's more convenient than playing it on the 7th fret of the A string and I won't hesitate to play an A on the 2nd fret of the G string instead of on the 7th fret of the D string. They are by far and away the best sounding strings I've used.
Prior to my current Jazz I had a 1965 Jazz Bass that I bought in 1966. It came with flatwounds (only thing available back then) and I replaced them once with a new set, and then while touring in the early 70's put on a set of the Fender Nylon Tape Wound strings. They were pretty dark sounding. I think Fender advertised that they had a "mellow sound" back then. I used those until 1976 when my '65 Jazz was stolen. So I played flat wounds/tape wounds for 10-years on that Jazz.
The flat wounds always sound darker than round wounds. That was the deal about round wounds when they became popular in the 70's, they had a much brighter sound because you could hear the higher harmonics, that were often not even there with flat wounds. RotoSound broke into the bass string market with their round wounds. The LaBella tape wounds are brighter than most normal flat wounds and yet are balanced better string to string than any round wounds I've used or heard. The semi-opaque White Tape ones are the brightest, the black are the darkest in sound, and the copper core and gold core are in between the two.
With the darker sound of the P-bass pickup, the white tape wound might make a difference for you. I actually prefer my '82 Jazz to the '65 I had. It has a slightly brighter sound because it uses the 70's era pickup spacing (4" center to center instead of the 3.6" spacing) which puts the bridge pickup closer to the bridge. It also has night and day better sustain than my '65 did. I attribute that to the all-maple neck on my '82. My '65 had maple with a rosewood fretboard. You really need those higher harmonics to bring out the more classic sound of a Jazz Bass. I'm not sure what you can do to get better sustain, outside of running a compressor on it.
AND, of course, it could be just like so many others here have said, that you've become attuned to the sound of your Stingray and Bongo, so nothing else will do. Trying a set of those LaBella tape wound strings should help answer that question and it's not a lot of money to find out.