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He was still the Red Sox's regular second baseman in 1963, but hit .234 in 143 games and lost his regular job to Felix Mantilla and Dalton Jones in 1964, both good hitters but mediocre-at-best fielders.
After the 1962 season, Green was traded to the New York Mets along with Tracy Stallard and Al Moran in exchange for Felix Mantilla.