Verrett spent his early years in The Woodlands, Texas, where he played Little League Baseball as an infielder.

As a relief pitcher in his freshman year, he recorded seven wins and a save in his first eight appearances, and served as the team's closer.

He moved into the team's starting rotation as a sophomore, and finished the year with a 5–3 win–loss record, a 3.28 ERA, and 97 strikeouts in 90+2⁄3 innings pitched.

[1] As a junior, Verrett compiled a 7–6 win–loss record, a 2.93 ERA, 96 strikeouts in 101+1⁄3 innings pitched, and had two complete games.

Between the two levels, he finished the year with a 5–2 win–loss record, a 2.70 ERA, and 93 strikeouts in 103 innings pitched, despite missing two months due to a strained rotator cuff.

[11] On December 11, 2014, during the Winter Meetings, the Baltimore Orioles selected Verrett from the Mets in the Rule 5 draft.

[16] Verrett pitched to a 0.73 ERA with 12 strikeouts in 12+1⁄3 innings before the Mets optioned him to Las Vegas due to the activation of Jenrry Mejía from the suspended list on July 7.

[17] The Mets promoted Verrett to the major leagues on August 18, when Bobby Parnell went on the disabled list.

[20] On April 13, 2016, Verrett made his first start of the season and second of his career against the Miami Marlins at Citi Field in place of Jacob deGrom.

[23] Verrett also recorded his first major league hit, a double on 1–0 pitch from the opposing pitcher Vince Velasquez to lead off the fifth inning.

[25] He made his Orioles debut on April 30, 2017, pitching two scoreless innings and getting the win against the New York Yankees.

On November 18, 2017, Verrett signed a one-year contract worth $800,000 with the NC Dinos of the KBO League.

[33] In 24 games (21 starts) for the Triple-A Rochester Red Wings, he compiled an 8-9 record and 3.80 ERA with 102 strikeouts across 125+2⁄3 innings pitched.