Those who came close, but fell short (1956-1989)
Since 1956, a number of Cardinals pitchers have produced great seasons but have ultimately fallen short of the Cy Young Award. The first Cardinal to make a serious run at the award was Ernie Broglio in 1960. Broglio led the league in wins and was the most valuable pitcher in the National League according to WAR, an advanced metric that attempts to calculate how many wins a player was worth in a given season. Ultimately, Broglio placed third, though he had a lower ERA than either of the two pitchers who finished ahead of him.
After Gibson won the award in 1970, the next Cardinal to finish in Cy Young Voting was reliever Al Hrabosky, who finished fifth in 1974. His teammate, starter Lynn McGlothen also placed, finishing ninth. Hrabosky would finish third the following year, behind Tom Seaver of the Mets and Randy Jones of the Padres. Though, in a strange turn of events, he finished ahead of both of them in MVP voting.
In 1977, former Cardinal, Steve Carlton, won his second Cy Young Award with Philadelphia. In a move the Cardinals came to regret almost immediately, they had severed ties with the promising young pitcher. He would win two more Cy Young Awards with the Phillies en route to induction into the Hall of Fame.
Legendary closer Bruce Sutter finished fifth in Cy Young voting in 1981. He'd won the award two years earlier as a member of the Chicago Cubs. In 1983 and 1985, Sutter again made runs at the Cy Young Award, though he finished third both times. Oddly enough, he finished second or better among pitchers in MVP voting in each of those three seasons. Sutter would spend four incredible seasons in St. Louis before moving on to Atlanta where he finished his career. He finished his career with exactly 300 saves and was inducted into the Hall of Fame. The Cardinals also retired his number.
During his five seasons in St. Louis, Joaquin Andujar produced a trio of top 10 finishes, including two top 5 finishes in 1984 and 1985. Andujar was known as a workhorse, and led the league in both innings pitched and batters faced in 1984, his best season in the majors. However, even at the zenith of Andujar's career, he wasn't even the ace on his own team. John Tudor was the best Cardinal pitcher in the mid-1980s. Tudor put together an excellent run with the Cardinals, and it all culminated in 1985. Tudor finished second in Cy Young Voting to Dwight Gooden. Tudor threw 275 innings and recorded an ERA of 1.93! Unfortunately, Gooden was sensational as well, and his 1.53 ERA season remains one of the greatest individual seasons of all time. In almost any other season, Tudor would certainly have captured the Cy Young Award.
Near the end of the 1980s, another young Cardinals starter enjoyed a productive stretch and placed in the top five of Cy Young voting. A 24-year-old Joe Magrane was one of baseball's best pitchers in 1988 and led the league in ERA. Unfortunately, he was unable to stay on the field and made just 24 starts, likely costing him significant recognition. He was sensational again in 1989, and this time finished fourth in Cy Young voting. Unfortunately, this was the best stretch of his career, and just several years later, at age 31, Magrane was out of baseball due to injuries and ineffectiveness.