In 1928, he ran for New York State Attorney General but was defeated by Republican Hamilton Ward, Jr.
Early in 1929, Governor Franklin D. Roosevelt appointed Conway as New York State Superintendent of Insurance.
In December 1939, he was appointed a judge of the New York Court of Appeals to fill the vacancy caused by the resignation[1] of John F. O'Brien.
In November 1940, he was elected on the Democratic and American Labor tickets to a full 14-year term.
He was elected on the Democratic, Republican and Liberal tickets Chief Judge in 1954, and retired from the bench at the end of 1959 when he reached the constitutional age limit of 70 years.