Five-digit zip codes were introduced by the U.S. Postal Service in 1963, along with two-letter state abbreviations. Who ever knew that ZIP stands for Zone Improvement Plan? The codes were only ever mandatory for second and third class and bulk mailing as of 1967.
The picture below in the center of Times Square was a temporary limited-service post office—note the dude at the window—and encouraged consumers to "Speed your military gifts and greetings," and "Mail early for Christmas," dating it around the time of the Vietnam War in the late 1960s. Incidentally, the zip code for Times Square: 10036.