On January 1, 2012, Jaye, along with Daniel Webb, was traded to the Chicago White Sox for Jason Frasor.

[2] On March 29, 2016, the Rangers traded Jaye alongside Bobby Wilson to the Detroit Tigers in exchange for Bryan Holaday.

[5] Jaye earned his first major league win in a September 5 game against the Kansas City Royals, pitching 2+1⁄3 scoreless innings in relief of starter Aníbal Sánchez.

[8] On May 29, 2018, the Cleveland Indians acquired Jaye from the Twins in exchange for cash considerations.

[9] In 9 games (5 starts) for the Triple–A Columbus Clippers, he struggled to a 2–5 record and 9.62 ERA with 16 strikeouts across 29 innings.