The next transaction listed on that page has him signing as a free agent with the Browns before the 1949 season...with no explanation as to how he gets there from independent San Francisco. The Feb. 16, 1949 Stanford Daily tells us what happened to Fine:
"Feb. 15 — The San Francisco Seals tonight announced outright release of pitcher Tommy Fine, who came to them last year from the Boston Red Sox."
So he was a free agent after being released in what the article inferred was a roster cut-down rather than a performance issue.
Fine made it back to the majors with the Browns in 1950, unfortunately with poor results again. He was optioned to San Antonio in the Browns' system, then to the White Sox' Memphis affiliate, still under option in 1952. Memphis then acquired Fine outright; this from the Schenectady Gazette of Feb. 15, 1953:
"The Browns also sold Tommy Fine, right-handed pitcher, outright to Memphis in the Southern Association. Fine was on option this year and played with Memphis the latter part of the season.”
Fine had a good year at Memphis, but was on the move again after the season. From the Montreal Gazette of Feb. 15, 1954:
“Richmond Virginians announced Saturday the purchase of Tommy Fine, veteran righthanded pitcher who last year played a key role in Memphis’ drive to a Southern Association pennant”
The article mentions a purchase from the White Sox.
Fine played into his forties with independent Memphis, then went to Shreveport for his final season.
Fine pic from the Baseball Birthdays site.