Oddly enough, much like with the modern country tone, a basic jazz guitar tone eluded me for a long time. Now, I fully admit that I don't tinker with knobs like I should. I tend to set the knobs on my amps, pedals, and guitar where I think they should be and then just leave them as opposed to tweaking knobs and listening to the results. This is probably why a basic jazz tone eluded me for so long.
Before I get into the specifics of finding a good jazz tone, let's look at what a good jazz tone is. Take a listen to this version of "How Deep Is Your Love" by Russell Malone and take careful note of where the guitar's tone fills up the tonal range.
You probably thought of words like "full" and "mellow" when you heard his tone. The difficult part for me to translate was that the tone was "full" in regards to the mid-range with balanced bass and treble. The hardest part for me to hear was that, in relation to how I typically prefer my guitars to sound for other styles, the tone on the guitar is rolled off (it's not on 10 or 11, if you're in Spinal Tap).
There are some essentials and then there are some nice-to-haves. You can get a good jazz tone with pretty much any guitar and you can get a better jazz tone with some great guitars, but don't feel like just because you have an Ibanez RG or a Tele that you sound jazzy.
First off, you need some decent strings. For less than $10, you can go a long way in improving your tone. I won't play on anything lighter than .10s and I only use that on my Tele for country twang. Ideally, for jazz, much like blues, the heavier the string, the fuller the tone. Try .11s, .12s, or (dare I say it?!?!) .13s. I prefer D'Addario XL EJ21 strings (.12, .16, .24w, .32, .42, .52; the w on the G string refers to it being a wrapped string as opposed to a plain string), although many players prefer flatwound strings. To me, flatwounds strings, while mellow, sound dead. I like a little life in my strings.
Now that you have some decent strings, make sure you're using the neck pickup. The further from the bridge (without crossing the mid-way point) the pickup is, the more mellow (less twangy, less treble) the tone will be. Also, make sure that you roll the tone back. That is, rather than running your guitar's tone knob wide open on 10, turn it back to 6 or so, depending on the potentiometer in your guitar's tone knob. You want it right at the spot where the highs are more subtle but the tone hasn't gone flat yet. It's a sweet spot that takes some time to find if you don't know what you are listening for.
So that gets you the basics with any guitar. If you really want to swing with the big cats, then you'll want to pick up a full-hollow-bodied guitar, often referred to as a "jazz box", like an Epiphone Emperor Regent. This guitar has a single floating humbucker in the neck position, which is ideal for a very mellow jazz tone. If a jazz box isn't an option for you, then try something a little more versatile, but still at least semi-hollow, like a Gibson ES-335 or similar model.
When it comes to the amp, it's pretty simple. You want a very clean tone. There are occasions when you might want some mild drive, but that's more for fusion. A good clean tube amp like a Fender Deluxe Reverb or Fender Twin Reverb will work great if you prefer a tube sound. Many jazz players prefer a solid-state amp like a Polytone Mini-Brute or a Roland JazzChorus-120. Solid-state amps work well for jazz because the warmth of the tone is coming from the guitar and the amp is just making it louder. Also, solid-state amps are lighter and require little maintenance, which is great for the gigging musician.
As for the settings, if you have a pre-volume or gain knob, make sure it is at the lowest setting possible so you don't get any distortion. Start with all your EQ knobs flat (that means 5-6 if your knobs are the typical 1-10 range, 11 for Spinal Tap) and make minor adjustments from there depending on your guitar and the venue. You don't need to be very loud because jazz isn't typically loud and if you really need to crank it, your amp is usually run through the house PA. An optional setting, depending on your taste, is to run a little reverb on your amp.
And that just about does it for the pedals. :) Well, kinda. The truth is that you really don't need any pedals for jazz. All the tone and sound comes from your guitar and your hands, which I'll get to next. There are a few nice-to-have pedals, but you don't really need any unless you're going to do a mixed set with some fusion or blues.
As with any setup, I like to run a Boss TU-2 Chromatic Tuner pedal for live tuning my guitars. I can't say enough good things about this pedal. It's fantastic and, yes, I know that sounds weird when I'm talking about what is essentially a utility pedal. Any pedal tuner will do and if you don't have a pedal tuner, just make sure you have easy access to a tuner of some kind.
I also like to run a volume pedal. This can help with hands-free fades both in and out of passages as well as volume swells for chord solos, etc. I like the Ernie Ball VP Jr.
If you're doing some ballads and you don't have any other chordal instruments (i.e. piano) to fill out the song, you might want to try a chorus pedal. I've used a few and they all have different flavors, so find one that you like. I have the Boss CE-5 Chrous Ensemble and the Visual Sound H2O Liquid Chorus & Echo. I like them both, although the H2O is a bit warmer, in my opinion.
Finally, if your amp doesn't have reverb, you might want to add a reverb pedal. I like the Line 6 Verbzilla, personally. It's really versatile and you can use it with many other styles as well from very subtle reverbs to massive, cavernous, ambient reverbs.
Again, any other pedals you might want to add would be to your personal taste, depending on what songs you are performing, but there really aren't any essential pedals for jazz.
Just like any style of music, the greatest impact on tone comes from your fingers, both right and left hands. You can affect your tone greatly through the precision, weight, and expressiveness of your hands. This is something that can't be taught (at least not from a web article) and must be learned through listening, mimicking, and practice, practice, practice. Even just thinking about the tone you want while playing can help translate this idea into your fingers and then into the guitar.
The only definitive piece of information I can share regarding tone and your playing is, just like using the neck pickup, you want to be away from the bridge to generate a warm, mellow tone. That is, your picking hand should be toward the neck rather than near the bridge.
Good luck with your search for a good jazz tone. If you've run across any secrets that you'd like to share, please feel free to comment or contact me directly and I'll add them to this article.