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All you have to do is look at the trim tag under the hood. It is riveted to the fire wall and the first line of information starts out with the word STYLE followed by 60 if it is a 1960 car. If you have any other set of numbers then check against the VIN tag riveted to the door jam the first digit should be 0 for the year 1960.
Here is the identification page from a 1963 Motor's m,anual to help figure out from the trim what year you have, though back then the cars changed sheet metal every two to three years with the first X frame the 1958 being an orphan with only one year sharing the same sheet metal.
Trim tags can be changed modified replaced if you so desire to clone a popular car but the VIN can not be touched. If it is altered or removed the car can be seized and crushed by the police.