Aged 16, Spittal made his Queen's Park debut away to Montrose on 17 November 2012. He went on to make a total of 18 appearances that season, scoring his only goal in a 5–3 win away to Elgin. The following season, he established himself as a first team regular, making 41 appearances and scoring 12 goals in the process. This form saw him pursued by both of Dundee's Scottish Premiership clubs. Having agreed terms with both sides, he opted to join former Queen's Park teammates Aidan Connolly and Andrew Robertson at Dundee United, over city rivals Dundee.
Dundee United
Having spent the first two games of the season as an unused substitute, Spittal made his debut from the bench in the club's third game of the season away to Celtic on 16 August 2014. Spittal played in the United squad that was relegated from the Premiership in 2016. He was part of the side that won the 2016–17 Scottish Challenge Cup as he played in the earlier third and fourth rounds plus the quarter-final, He was injured for the semi-finals and final however.
Partick Thistle
Spittal joined Scottish Premiership club Partick Thistle on a two-year contract on 15 June 2017. He scored three goals in his first three games for the club, scoring twice against St Mirren in the Scottish League Cup, including a free kick. He then scored in a 2-1 away win against Airdrieonians in the same competition. Spittal marked his league debut for Thistle with 3 assists against Aberdeen in a 4-3 defeat. In his second league appearance for the club, he scored his first league goal away to Ross County in a 1–1 draw. Spittal continued his form the next week with a goal direct from a free kick and another assist as ten-man Thistle drew 2–2 with Rangers at Firhill, taking his tally to 7 appearances, 5 goals and 4 assists in all competitions. He then carried his form into the next League game opening the scoring in a 1–1 draw with Hearts. He scored his fourth league goal of the campaign with another freekick, scoring the opener in a 3-2 home win against Motherwell. On 26 February 2019, in the absence of Stuart Bannigan, Spittal took on the captain’s responsibilities as Partick Thistle travelled to Dunfermline, They lost the game 3–0. Spittal again lead the team out in a home game against Hearts in the Scottish Cup Quarter Final which ended 1-1 meaning there would be a replay in the capital. Spittal finished the season with 8 goals in all competitions and was awarded the club's Player of the Year award. Spittal informed the club that he would be leaving on the 31 May 2019. In all competitions, he made 89 appearances for the Jags, scoring 15 goals and registering 13 assists.
Ross County
On 4 June 2019, Spittal signed for Ross County, agreeing a two-year contract. He made his debut in the league cupat home to Montrose providing an assist. Three days later he scored his first goal for the club during a 4-0 away win against Brechin City. Only five days later he provided another assist during a 2-1 away win against group favourites St Johnstone. Ross County would finish top of their league cup group with twelve out of twelve points and advance to round two. Spittal would start his first league game on the bench but was subbed on after only 11 minutes to help his side defeat Hamilton 3-0.
