The name "Jet Propulsion Laboratory" dates to 1943. They do rockets. At least 2 years prior to that name change they had worked on JATO, which is "jet-assisted takeoff". The company Aerojet was founded in 1942, making solid-fuel rockets.
Like many terms in the English language, the word "jet" has more than one meaning. In the context of flight above the surface of the Earth, it refers to an oxygen-breathing device.