It’s an anachronism that probably goes back to bankers and accountants wearing green eyeshades and having green shades on their desk lamps. For people who had to use their eyes constantly (all day) in situations where a small numerical error could cost their company a lot of money, green was generally accepted to be the most “restful” color for eyes and supposedly reduced eyestrain in harsh lighting conditions (factories, offices, etc.— and daytime baseball games).
Green was the original color for sunglass lenses and is still one of the best choices for easing the load on the eyes while not distorting vision.
For the same reason players wear eyeblack under their eyes (sunlight reflecting off sweaty or shiny faces), the visor of a baseball cap has to reduce glare as much as possible. Squinting for nine innings is unpleasant.
Because bankers and accountants started using these in the later 19th century, it stands to reason that green would be the first color used on the underside of a baseball cap/visor. But as another answerer pointed out, today most caps have grey undersides on the bill.
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