Moon Pilot (1962)

"A funny thing happened on the way to the moon…"

An Air Force captain inadvertently volunteers to make the first manned flight around the moon. He immediately falls under the watchful protection of various security agencies, but despite all their precautions, a young woman who may be an enemy spy succeeds in making contact with the captain. The captain eventually discovers that this woman is not an enemy but rather a friend from a very unusual source.