If you want the woody sound, getting a (semi) hollow body archtop helps a lot!
But for "just" jazzy sounds, I think a strat/tele solid body can do the job very well too.
Example: Ted Greene the great master @
http://www.youtube.com/watch?v=zDuee6blvj8
[Cheap solutions to get closer to that sound]
1) Thicker pick. I like Jim Dunlop Jazz III (red color)
2) Thicker strings and/or flat wound.
[FREE, but requires training]
2) Dig deeper (not harder, though that gets you a different dynamic that's useful as well) when picking to get fatter sound. As opposed to just using the very edge of the pick, which gets sharper/thinner sound, which is useful too. I like the variety. Just by varying my pick depth can change the tone.
3) Play closer to the neck. (Generally: Neck rounder warmer, Bridge thinner brighter)
4) Use your fingers (meat, not nails)
5) Use your thumb. (ala Wes Montgomery)
I bought Pacifica 112J as my first electric and found the sound wanting and could not for my life as a total newbie get any nice sounds out of it. After I got a Epiphone archtop (Joe pass) the sound is just right there, no effort. Though I had to learn to play it right, as over time I noticed that different type of electric guitar requires different picking technique.
Archtops have to be picked softer (when amped) and with more follow through (like stroking it) to get that typical jazzy woody smokey sound. The soft unplugged sound is lovely too.
Solid bodies (I only play clean with some reverb, no experience with drive/distortion), to get it to sing also took me very long. Going back to it every few months trying to coax a nice clean sparkly fender sound out of it. Took a period of 5 years (since I busy at work and play the guitar rarely) to get a tone I can actually sit down and practice with, since the ugly sound is just a major turn off for me. So my patience has paid off, as I wanted to sell the guitar off at the beginning thinking it's the guitar... though a better guitar with actually decent pickups would be a good upgrade after acquiring the skills on a "training" guitar. (I'm still training..
)
Sorry if that looks like a wall-of-text attack, but I identify with the situation and trying to help as much as I can.