He was battling injuries late in his career and bounced from the Dodgers to the Tigers, then the Yankees. He has a year off from the majors, then appears once more for Brooklyn.
His Baseball Reference minor league page fills in the 1955 gap: he was in Minneapolis in the GIANTS organization. Yes, even with his history with the Giants.
The Giants and Minneapolis had been after him for awhile: they had offered him a contract in 1954 before he signed with the Yankees. From the Evening Day of New London CT July 23, 1954:
“Branca…rejected an offer by the New York Giants to hurl for their Minneapolis farm club in the American association.”
The Yankees signed Branca after Detroit had released him. His ERA was under 3.00 in his short time in the pinstripes, but he walked more than one batter per inning pitched and was let go. This time, he took the Minneapolis offer. From the Pittsburgh Press of Dec. 10, 1954:
“Branca…was signed as a free agent yesterday by the Giants’ Minneapolis farm team of the American Association”
He was hoping to catch on with the Giants, but it wasn't to be. A 3-3 mark and an ERA over 7.50 got him released, according to his book and this article from the Troy NY Times Record of March 5, 1957:
“I haven’t done much pitching since I was release by Minneapolis in July, 1955…”
The article is about Branca in spring training with the Dodgers, who had signed him in 1956. At the end of that season, they took him off the roster in this move, chronicled in the Daytona Beach Morning Journal of Oct. 18, 1956:
“Branca…was sold to St. Paul of the American Assn.”
Branca never played for St. Paul, he retired in the 1957 spring training though there are newspaper reports that he had thrown pretty well at that time.