May spent the most time with the Angels, but it was with contending Yankees and Orioles clubs that his contributions gained the most attention. He had only one winning season with the Angels, 1972, when he was 12-11 with a 2.94 ERA. However, he showed good skills, and the Yankees, looking to start contending again under new owner George Steinbrenner, purchased May in June 1974.
May went 8-4 for second-place New York and finished the year with a 3.19 ERA, then went 14-12 in 1975. In June 1976 he was part of a blockbuster 10-player deal that sent him with Tippy Martinez, Scott McGregor, and Rick Dempsey to the Orioles. He went 11-7 the rest of the way and 18-14 in 1977, and was traded with Bryn Smith in the six-player deal that brought Don Stanhouse from the Expos. A slump to 8-10 in 1978 led to a switch to relief in 1979; he went 10-3 with a 2.30 ERA in both roles that year.
May returned to the Yankees as a free agent after that season and had his best year in 1980. Used as a starter and a reliever, he led the league with a 2.47 ERA and went 15-5, second in winning percentage behind Cy Young Award winner Steve Stone. He took a complete-game, 3-2 loss in Game Two of the LCS against the Royals. May was less effective thereafter.