He arrives from Scottish outfit Edinburgh on a three-year deal after it was announced that France's Jaminet would leave Toulouse for league rivals Toulon over the coming weeks.
Kinghorn can also feature on the wing and at fly-half, helping Ugo Mola to deal with a lack of depth in the number 10 shirt.
First-choice playmaker Romain Ntamack is out until at least next year with a serious knee injury with full-back Thomas Ramos and scrum-half Antoine Dupont stepping into the role.
Les Bleus captain Dupont will also be unavailable for Toulouse during the season as he prepares to play seven-a-side rugby at the Olympic Games in Paris.
Kinghorn and Jaminet's mid-season moves are unusual in rugby with players preferring to respect their contracts, which normally finish at the end of campaigns.
South Africa's two-time Rugby World Cup winner Cheslin Kolbe also changed clubs during a season, going from Toulouse to Toulon like Jaminet in 2021, for a reported fee of more than one million euros (£870,000).
"We agreed reluctantly to this release in return for payment of a significant sum from Toulouse, which we'll now consider carefully how best it can be reinvested in the club," Edinburgh coach Sean Everitt said in a statement with the outfit saying it was a six-figure fee.
"We wish Blair all the very best in his new challenge in France and thank him for his commitment to the club," he added.
Kinghorn will arrive in Toulouse on December 4th and could make his debut against Cardiff in the Champions Cup five days later.
