The very first electric guitars were basically acoustic guitars with a electronic pickup. Pioneered by Eddy Durham in the 1930's. One of the first models: Gibson's ES-150.
Use a hollowbody guitar for a round, warm Jazz sound. Together with a humbucker pickup and flatwound strings, brings you to smokey bars of the 1940's and 1950's.
Player's well known with a “hollow” sound are Charlie Christian, George Benson, Wes Montgomery, Joe Pass and many more. For a more Rockabilly sound, look at Gretsch.