Mary Landrieu
was a close third in the LA Governor primary in 1995 and was elected to the Senate in 1996
(which also featured a very competitive jungle primary).Similar with Susan Collins; she was third in the ME Gov race in 1994, the bounced back in 1996 to win her Senate seat.
Even Huey Long
lost his first Gubernatorial primary in 1924 before winning in 1928.
Then there's the two more recent examples of Pat McCrory and Bobby Jindal. They were actually
really similar. Both lost close races to the first female Governors of their state, then were elected easily after the women they beat retired after one term.