It actually looks like an early Gretsch, but that doesn't quite line up with their historical timeline for electrics. They had a shift toward electric guitars starting in 1942.
In addition, while with Glenn Miller Bobby Hackett had been asked to buy an amp and ended up with one, but reportedly never actually plugged it in.
So, it is possible that this may have been an acoustic guitar - I have not been able to nail it down.
I can share that when I was a kid, way back in the early 60's, we went on a family vacation to Disneyland, and while there we all got to watch a live concert by the band of Tex Banakee (don't know correct spelling) - who was in Glenn Miller's orchestra. They did all of the Glenn Miller stuff, with the stand-up and sit-down horn playing and the whole bit - WAY kewl. I have never forgotten how cool it was to have been able to watch someone from that big band era playing live.
Sorry I couldn't give you more info - maybe someone else can figure this one out for you.
Bob Bone