I can think of one more manager in recent years whose fortunes turned after a postseason series loss to Detroit.
A's manager Bob Melvin.
The A's actually fell in consecutive ALDS game 5's to the Tigers, worse yet both at home. At the time I thought the second time was enough to get Melvin fired, especially given the A's previously had a 2 games to 1 lead. Melvin was retained, a decision the A's would come to regret. They regressed from 96 wins to 88 in 2014, and could only manage the second wild card berth that year. They lost to Kansas City in the wild card game. They regressed even further in 2015, finishing with only 68 wins and in last place. Though they improved to 69 this year, they still finished last. You think Melvin would be fired now.
Just goes to show you, you do NOT want to lose to the Tigers in the postseason. Your fortunes as a manager will turn on such a series loss. Joe Girardi and Bob Melvin have learned that the hard way.
If Melvin does get fired, I would like to see former A's star Rickey Henderson try his hand at managing. He stole more bases in his first 13 seasons than Lou Brock did his whole career - and Henderson didn't stop stealing bases until his career was over.