How do I get a "jazz" like tone on my guitar?

With your guitar and amp, you are going to want to drop the highs; I suggest you use the "1" or "2" pickup switch position. You want the neck pickup or the neck and middle. Roll the tone back to 1-4 on the Strat; depending on how much high end you want coming through. If you are playing older style big band, and just comping chords, you won't need much in the way of Drive or Gain. You want a mostly clean tone, but like I said, run it all at mid levels, and roll down any spurious highs to get a mellow tone.