No, it’s the other way around.
Reid served as lieutenant governor from 1971 until 1974, when he ran for the U.S. Senate seat being vacated by Alan Bible. He lost by fewer than 700 votes to former governor Paul Laxalt.
It’s curious that Paul Laxalt chose not to run for another term in 1970 as Governor. I think he was getting tired of Howard Hughes always calling him at all hours of the night.