Getting a good jazz tone may have very little to do with such things as string types or amp brands.
While those things can help put you in that ballpark, the very same sounds that you seek could very well be in that same guitar that you're cradling now - and you don't even have to change anything about it at all.
Of course, we've all heard that talk about flats and all that jazz, but I wouldn't consider them to be a 'necessity' in getting a good jazz tone.
I say that because I've gone through that same phase years ago. I used to have that notion that flats were the ONLY way to go to get that jazz tone going.
I even went as far as putting .012 D'Addario Chromes on my Yamaha 335 - thinking that that would make my tone even 'jazzier' ..
To cut the long story short, I took 'em off right away. The tension was just too much for the guitar, and I didn't want to tweak with the guitar's neck just to accommodate the new strings. Besides, the flats felt super stiff on the 335 then. Sure, it sounded jazzy, but I wouldn't want to sacrifice the guitar's playability for it.
Somehow, I still craved for the feel that the normal .010 roundwounds offered. I felt more confident knowing you could bend a note anytime.
Not that I'm saying flats aren't great. Don't get me wrong.
Let's just say that if you do decide to try them, just be prepared to deal with the extra tension, and the other tradeoffs such as not being able to bend fully up a whole step or more (unless you're SRV).
Which brings us back to roundwounds ... and Jeff Golub, as extreme mentioned earlier.
It's interesting to note how a smooth jazz guitar player like him uses ... (guess what?) .09s on most of his guitars. He said so in the Gear section of his webpage, citing his tendonitis as the main reason behind it.
But see, even with such a gauge, the man sounds as jazzy as jazzy can be.
I think getting a good jazz tone has a lot more to do with getting acquainted with the guitar's volume control. From experience, you can get great jazz tones with a humbucker-equipped guitar (like a 335 or even a Les Paul), with a guitar's volume around 5 only. That way, the sound isn't as punchy, and the attack isn't as pronounced compared to when it's on full volume.
Heck, you can even dial the same tones on a Strat or a Tele.
Lastly, maybe it would also help to use thick picks
So there .. hope that helped