York only missed out on the runners-up spot in 1958–59 on goal average, and were promoted for the first time in third place. However, they were relegated from the Third Division after just one season in 1959–60. York's best run in the League Cup came in 1961–62, the competition's second season, after reaching the fifth round. They were beaten 2–1 by divisional rivals Rochdale. York had to apply for re-election for the second time after finishing 22nd in 1963–64, but achieved a second promotion the next season, again in third place in the Fourth Division. York were again relegated after one season, finishing bottom of the Third Division in 1965–66. The club was forced to apply for re-election in three successive seasons, from 1966–67 to 1968–69, after finishing in the bottom four of the Fourth Division in each of those season. York's record of earning promotion every six years was maintained in 1970–71, with a fourth-place finish in the Fourth Division.