I love jazz guitars... I own several and they do just what you described quite well, but at higher volumes the feedback can be a bit hard to control.
If you want something that sounds good unplugged, I strongly recommend a spruce top. The flame-maple top semi-hollow guitars sound GREAT plugged in, but they don't have the acoustic tone of a spruce-topped guitar.
My personal favorites are Epiphones. The AlleyKat is one of the best semi-hollow guitars for the buck... small and easy to handle but packs a lot of tonal punch for the size and money...
The Epiphone Joe Pass has a really nice resonant spruce top. Mine is modified with Gibson 57 Classic pickups and it sounds great from acoustic to high-gain...
The Dean Palomino packs a really great set of P-90's... 3 pickups with a 5-way switch and the guitar sounds very nice... they can be had cheaply on the used market...
Took a moment to locate this pic... but if you are in the market for something to be played mostly plugged in and willing to foot a few extra bucks, you'll find the Hamer Artist Korina is hard to beat for right around the $1K mark...