Cole came through Tranmere Rovers's youth system. He made his Football League debut against Rochdale on 17th January 2012 as a late substitute. Tranmere awarded him his first professional contract in April 2012.
On 15 August 2012 Cole scored on his first start for the Prenton Park club against League Two side Chesterfield in the League Cup, a game which Rovers won 2–1. His first Football League goal for the club came on 7th November in a victory over Hartlepool. He scored again against Chesterfield in the FA Cup second round, as Tranmere knocked Spireites out of the second cup competition of the year.
With one year left until his contract expired, Cole extended it until summer 2017 on 26 August 2014. Four days later he scored two goals against Morecambe as Tranmere won the game 2–1. He made ten straight appearances at the start of the season, but later was ruled out of play for two months after picking up an injury before the game against Bury. On 3rd January, Cole scored a late goal against Premier League side Swansea City as Tranmere lost 2–6 and were knocked out of FA Cup.
After a short loan spell at Morecambe in March 2016, Cole signed for Scottish club Hearts in June 2017. Stockton made his first appearance for Hearts in a friendly against Livingston FC scoring the only goal, as Hearts won 1-0.
In August 2018 Cole returned to Prenton Park, signing a one-year deal with the club, making more than 100 appearances over 2 spells.