As long as a guitar feels good and has a good clean sound, imo that's all you need to play jazz. The only reason to get caught up in what is "right" is if you're playing out to audiences that require the right "look".... A lot of hardcore rockabillies, for example, won't go and see a band that plays very modern look equipment as they won't have the right look - very much part of the package. Happens all over the musical spectrum - audiences expect the look as much as anything. Branding / marketing aside, I say use whatever you like. Sometimes some resl creativity can spirng from using the "wrong" guitar...