Baddoo, 24, suffered a right quad strain during the fourth inning of the Tigers' 11-6 loss to the Arizona Diamondbacks on Friday.
He is batting .231 with three homers and 16 RBIs in 50 games this season.
Solak was claimed off waivers from the Atlanta Braves on Friday. He was then optioned to Toledo after Detroit cleared room on the 40-man roster by transferring right-hander Spencer Turnbull to the 60-day injured list.
Solak, 28, was designated for assignment Tuesday after appearing in just one game for Atlanta. He is a career .252 hitter with 21 homers and 93 RBIs in 254 games with the Texas Rangers (2019-22) and Braves.