Morstead began his football career at Southern Methodist University, where he played as a punter for the college team. In 2009, he was drafted by the New Orleans Saints in the fifth round of the NFL Draft. Morstead quickly made a name for himself in the league with his powerful punts and precise accuracy. He spent 12 seasons with the Saints, becoming a critical part of the team’s special teams unit and earning numerous accolades for his performance on the field.