Now is probably the best time in all of history to be looking for a well-made semi-hollow for 1K or under...

I have a great Hamer Echotone (ES-335) I bought barely used for $200, from a jazz guitarist who had the frets nicely dressed and had it set up with flatwound strings with a wound third (the key to old school "jazz tone") and had the pickups swapped out for Seymour Duncan '59s already. Killer.

I also bought a little Epiphone Casino Coupe (fully hollow) barely used for $200... it needs some fret work and I'm swapping out the pickups (overwound because that seems to be what "the kids" expect, which robs the guitar of brightness and clarity) for some GFS P90s (such a simple pickup that I don't hear much difference between the cheapest and the most expensive boutique versions as long as they're wound in the right lower output range).

Epiphone also has a version of the ES-339 (these and the Casino Coupes have smaller Les Paul sized bodies).

I think there's some semi-hollow Epiphone Les Pauls around (and let's not forget that the solid Les Paul was designed and intended to be a jazz guitar!).

The Epiphone Dots are good.

The Ibanez Artcores are good.

You might be able to score a Gibson Mid-Town or even an ES-135 used for under $1K (I've seen them go for $800, often).

There's the Es-335 Studio, and the ES-333 (a 335 that was cheaper because it had a back panel to assemble/access the electronics, which I and the rest of the world probably find preferable) that was in production for a short while and people who have them don't seem to part with...

and then, stepping away from the "purist" angle...

I've heard amazing jazz guitar tone floating out of restaurants in the French Quarter in New Orleans, only to look in and see someone playing a Steinberger guitar, or a Parker Fly, or a Line 6 Variax...