The 23-year-old, who captained Oldham Athletic last season, has signed for the Robins for an undisclosed fee.
Smith signed for Latics initially on loan in 2013, before signing permanently during last summer.
The midfielder made over 50 appearances for the club, scoring two goals. His strike against Leyton Orient towards the end of last season was voted the 2013/14 Oldham College Goal of the Season.
Oldham Athletic would like to thank Korey for his efforts and wish him luck for the future.