In 1976, he broke Hoyt Wilhelm's American League record of 154 career saves, and the following year eclipsed Perranoski's major league mark for left-handers of 179 career saves. Through 1977, Lyle had compiled 201 career saves, and was within range of Wilhelm's career big-league record of 227. Lyle was associated with a trademark song to herald his entry into games, "Pomp and Circumstance March No. 1 in D". But despite Lyle's 1977 Cy Young Award, the Yankees decided to upgrade the closer position in the 1977 off-season. To this end they signed Pittsburgh Pirates' reliever Rich Gossage, who had been an All-Star the previous three years and had led the American League in saves while with the Chicago White Sox in 1975. Gossage was a power pitcher as opposed to Lyle's more slider-based approach and could throw his fastball much harder than Lyle could, at or near 100 MPH on a consistent basis. Although Lyle was not opposed to the Yankees' move, he later said that they should have gotten rid of him after they signed Gossage. Lyle knew the 26-year-old Gossage was a rising star who would eventually become the go-to reliever, and Lyle was not okay with a reduced role in the Yankees' bullpen. During the 1978 season, working mostly in middle relief appearances, Lyle appeared in 59 games and recorded nine saves, most of which were early in the season. He also posted a 9-3 record out of the bullpen, but posted his second-highest earned run average. Lyle was left off team's postseason roster after suffering a varicose vein flareup in his pelvic area. Yankees teammate Graig Nettles quipped that Lyle went "from Cy Young to sayonara." During the course of the season, which as noted he documented in his book, Lyle expressed a desire to leave the Yankees. One of Lyle's desired destinations was the Texas Rangers, and he believed that a deal would be done "before Christmas". On November 10, 1978, the Yankees traded him to the Rangers along with catcher Mike Heath, pitchers Larry McCall and Dave Rajsich, and minor league shortstop Domingo Ramos in exchange for outfielder Juan Beníquez, pitchers Mike Griffin, Paul Mirabella, and Dave Righetti, and minor league outfielder Greg Jemison. $400,000 in cash was paid to the Rangers. Lyle signed a new contract with the Rangers shortly after he was acquired, with the intention of him finishing his playing career in Texas and being paid $50,000 for each of ten years as an analyst on Rangers telecasts.