The circuit that Leo designed around 1952 or so has the tones you're looking for. There's a million variables out there, I like to stick with basics.
Basics like actual Telecaster pickups. Endless tone searching with pickups I'm glad I never got into. Play what's in the guitar and make it work by adjusting the knobs and the amp, like Roy Buchanan, Albert Collins, Danny Gatton, Mike Bloomfield and countless other who played these guitars as is, before a multi million dollar aftermarket pickup industry was invented. The only real departure from the norm were a couple players, Collins, Richards, who put a PAF in the neck position. Fender was paying attention and got Seth Lover to design a "full range humbucker" for the Tele, unveiled in 1972.
The original 1952 Tele circuit goes like this.
Position 1) Bridge pickup only; Bright, cutting, steely. This is largely thanks to a massive ferrous bridge plate with brass saddles, beefy angled pickup. The tone pot works as normal in this position with a very effective 0.047uF tone cap.
Position 2) Neck pickup only; Sounds like pure blues, round, full. The tone pot works the same is in position #1.
Position 3) Neck pickup through a very wooly sounding - 0.1uF - tone cap. Instant Jazz tones, you can even play a little bass on the low strings. The tone knob does nothing in this position.
You can get the switch to hang between position 1-2 giving you an out of phase mix of the the two pickups. Something they later incorporated into the Strat, but not until 1977!
Already by 1954, the basics were too basic and they unveiled the Strat. The Strat doesn't have the same tones at all.
Not as bright and steely on the bridge, the middle makes a good rhythm tone, the neck gets a good blues tone.
The out of phase positions are "quacky", and with a reverse wound middle pickup the two positions become hum canceling, this is a big deal for recording, needing to be quiet between songs, etc.
A great player can get a great tone out of either a Strat or a Tele, but of the two I think the Tele is best suited to jazz.
Not only this is true, but there's anecdotal evidence that "great players" can get "their tone" out of any guitar/amp. I remember reading an article about Clapton where he plugs in to some crappy guitar and amp at a house party or something like that, he "twiddled a couple knobs" and voila, Eric Clapton.
When Jazz went wild...
In this video he channels 70's Jeff Beck Blow by Blow tones, before morphing into George Benson, and back to Jeff on an ancient Telecaster that in his words; "I got from Danny Gatton. Before that it was Roy Buchanan’s. It was a great guitar that was stolen when somebody pulled a gun on me in Boston."