Conway was the 30th mayor of New Orleans (June 10, 1868 – April 4, 1870). In 1862, he moved to New Orleans and began work at a cotton commission house (which buys and sells cotton for a customer), but was soon fired due to the suspension of business in New Orleans by the occupying Union forces. Foreign quite a while, Conway had been greatly interested in politics, and considered active participation as part of his patriotic duty.
After the Civil War, when the Orleans Parish Democratic Committee was reformed, Conway was named chairman.
Conway worked with the Democratic State Central Committee to return control of the city back to the people, after several years of military control. Also during Conway"s time as mayor, the city of New Orleans received a statue of Benjamin Franklin from neoclassical sculptor Hiram Powers.