Moon Pilot (1962)

After Capt. Richmond Talbot (Tom Tryon) is selected to be the first astronaut on the moon, the U.S. government begins shadowing him to ensure that enemy operatives don't learn about his mission. Richard's superiors monitor his every move and become suspicious when alluring foreigner Lyrae (Dany Saval) begins to follow him. Richard knows that the mysterious woman means well, but he's a bit surprised to find out she hails not from a different country -- but from another galaxy altogether.