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Morris made his professional debut for Leyton Orient in October 2001, in the 2007–2008 season he broke into the first team and made his 100th appearance (including cup matches) for the club. On 5 August 2010, Morris signed a one-year deal with Southend United. Morris was given the number 1 jersey and established himself as Southend's first choice goalkeeper. After 37 appearances in all competitions, Morris was rewarded for his performance with a further one-year contract. On 1 August 2012, Morris played for Portsmouth in a friendly at Aldershot Town and then signed for Aldershot on a six-month deal. After making 6 appearances during the 2012–13 season, Morris signed a one-year contract extension. In 2014, he signed for Gillingham on a one-year deal. He signed a one-year extension at the end of the 2014–15 season, but was released at the end of his contract. After leaving Gillingham, he joined Crawley Town in May 2016 as a player-coach, on a one-year contract. He won the Player of the Year award for Crawley Town for the 2016–17 season. During the 2017–2018 season, he made 44 league appearances for the club and won Crawley's Player of the Year award for the second time. before signing a contract extension in January 2018 until the summer of 2020. He appeared in all 46 of Crawley's league games during 2018–19 and received a trio of awards for his performances that season, again receiving the Player of the Year award, alongside the Players' Player accolade and the award for the best away player. In September 2019 he signed a new contract until 2021. At the end of his contract, the club extended it for another year. In 2021–22 he was named as the side's Player of the Season for a fourth time. In July 2022, Morris returned to Gillingham on loan until January 2023. Shortly after his loan spell came to an end, he agreed to terminate his contract with Crawley and join Gillingham on a permanent basis. In April 2023 Gillingham confirmed that he had signed a new contract but did not specify its length.