Conway's Baseball-Reference page lists the deal that sent Conway to New York, but nothing afterwards, and he did quite a bit of traveling.
The Giants sold Conway from their Minneapolis AAA team in the 1948-49 offseason. From the Pittsburgh Post-Gazette of Dec. 21, 1948:
“The Indianapolis American Association baseball club has purchased Infielder Jack Conway from the New York Giants.”
Conway hit a career-high 13 homers for Pittsburgh's AAA club, then moved of to San Francisco the next year. From the Milwaukee Journal of Apr. 20, 1950:
“The Indianapolis Indians of the American Association Wednesday night announced the sale of shortstop Jack Conway to San Francisco of the Pacific Coast league.”
San Francisco was independent in 1950; in 1951 they were a Yankees farm team and moved Conway again right after the season started. From the Toledo Blade of Apr. 6 1951:
“The [Toledo Mud] Hens also announced the purchase of Jack Conway, a utility infielder from the San Francisco Seals.”
He stayed in Toledo (Tigers affiliate) for one season, threatening to retire at the end, but went to Ottawa (Athletics affiliate) just before the 1952 season in a deal I could not find. He hit less than .200 there for the year and was given an option to coach somewhere, but instead went to Shreveport. His playing days ended shortly thereafter.