The 20-year-old, who has been with the Robins for four years, signed his first professional deal in 2015.
The former Hereford loanee missed much of the 2016-17 term with a knee injury.
"I want to make up for the last couple of years and get back playing. I am very grateful to the gaffer and the medical team. I really want to repay them now," Page told the club website.
Cheltenham Town winger Adam Page has signed a new one-year contract with the League Two club.