Gigot spent time in the youth team at London Broncos. He signed with Catalans Dragons in 2010 and made his Super League debut that year, playing a total of 15 games and kicking two goals. He also represented Catalans in the 2010 Challenge Cup where he played 2 games and scored 1 try. In 2011, Gigot played 7 games throughout the season for the Catalans Dragons, kicking 1 goal. He once again represented Catalans in the 2011 Challenge Cup, where he played 1 game. As well as playing for the Catalans Dragons, in 2011 Gigot played 9 games on loan for Toulouse Olympique in the 2011 Championship. He scored 5 tries. After being released by Catalans, Gigot signed with the Avignon Bisons to play in the 2012-2013 Elite One Championship.
Cronulla Sharks
Gigot was given an early release by Avignon at the start of 2013 so he could travel to Australia for a chance to play in the Australian National Rugby League competition. Gigot spent pre-season training with St. George Illawarra Dragons before signing a one-year deal with the Cronulla-Sutherland Sharks. He did not feature for the Sharks at NRL level, however.
London Broncos
In February 2014, Super League club London Broncos announced the signing of Tony Gigot for the 2014 season. He played two matches and kicked four goals before leaving the club to re-sign with Avignon.
Return to Catalans
In May 2015, the Catalans Dragons announced the resigning of Gigot. He made his second debut for the club playing as a centre in a 37-34 victory over Featherstone Rovers in the Challenge Cup. The following week he made his Super League return for the Dragons, scoring a hat-trick in a 58-14 victory over Wigan Warriors. On 10 February 2017 it was announced that Gigot had been banned for two years by the French rugby league authorities for an incident with an anti-doping official during a training camp in October 2016. It was reported that Gigot had not failed a drug test and that supported by his club, Gigot will appeal the ban. An appeal against the length of the ban was successful and with the ban reduced to three months, Gigot was able to play again from April 2017. However in August 2017 the full two year ban was reinstated by the French anti-doping agency for the "inappropriate exchange" with the anti-doping official. Gigot and the club launched a further appeal which was successful and on 9 February 2018 it was announced that Gigot was free to resume playing. On 5 August 2018 Gigot was named Man of the Match in Catalans' 35–16 victory over St. Helens in the semi-final of the Challenge Cup. At the end of the final on 25 August he was named as the winner of the Lance Todd Trophy, becoming the first French player to win the trophy as Catalan beat Warrington Wolves 20–14.
At the end of 2019, Gigot and Catlans Dragons failed to come to terms on a new contract. As a result, Gigot started the 2020 Super League season without a team and trained with Avignon to stay fit. In February 2020, the Toronto Wolfpack announced that Gigot would join the team for a four-week, unpaid trial period.
