The Canadian international, 24, joined Glasgow Warriors for the 2014/15 season, initially on a three-month deal, and has made seven appearances for the club, accumulating 11 points with four conversions and a penalty.
Connor, who went on loan to London Scottish in December 2014, returned to Glasgow in February and was involved with the club during the Six Nations.
He told glasgowwarriors.org: "I would like to thank everyone at Glasgow Warriors for giving me the opportunity to play and be involved at this great club.
"I have learned a lot in my time here, and will use my experiences as I look to progress in my career.”
Glasgow Warriors head coach Gregor Townsend commented: "We've seen Connor (Braid) develop hugely over his short spell here and at London Scottish.
"We've called upon his services a number of times throughout the season and he has always stepped up to the plate.
"He is a great person to work with and we hope he continues to prosper and shine wherever he goes."