He went on to the White Sox and injured his arm, and then bounced around in the minors for a couple years. In 1959 he spent the whole year at AAA Indianapolis, but then it becomes confusing.
Qualters' Baseball Reference minor league page has him playing for three teams in 1960.
This SABR article on Phillies' bonus babies has Qualters doing the following in 1960:
"Qualters then began the nomad stage of his career, bouncing from Indianapolis to Houston and then San Diego in 1960 (7–9, 4.47 combined)"
Which would make sense, given that he played for Indy for all of 1959, logically he would have started there in 1960. But it ain't so...
The Terre Haute Star of July 13, 1960 has the following:
"INDIANAPOLIS, July 12.—(UPI) —The Indianapolis Indians today announced the signing of a new pitcher as Manager Ted Beard attempted to push his sagging team from the American Association cellar. Tom Qualters, 25 years old a right-hander, was signed as a free agent. He was released by the Houston team, where he had a 3-2 record and an earned run average of 5.57, appearing in nine games in relief."
So he didn't start the season at Indy. And we find in this article from the Salt Lake Tribune of April 10, 1960, that he didn't start the season in Houston, either:
"INDIO, CALIF., April 9 (UPI)—Righthander Tom Qualters halted a four-game losing streak for the San Diego Padres here Saturday by stopping the Portland Beavers, "
So the correct timeline is San Diego, then Houston, then Indianapolis. We don't know how he got to San Diego or Houston yet, but we at least know the order.
Various sources indicate that Qualters started 1961 in Indianapolis, then went to Dallas, tried to retire, and went instead to single A Williamsport to player-coach for awhile before finally cashing it in.
So we have a lot of questions to resolve about Qualters' last two seasons!