Zack Greinke
I'm starting off this list with the two least likely options. Zack Greinke is only marginally more likely than Tink Hence, if that. Greinke is currently a free agent, and he's been working out in Arizona to stay as fresh as can be should a call come his way.
Greinke absolutely fits the age minimum to be a starting pitcher on the Cardinals in 2024, as he is 40 and will remain so until late October. When the 2023 season ended, Greinke wasn't quite sure if he would return for another season. The 20-year veteran has already racked up quite a few accolades including a Cy Young Award, 6 Gold Gloves, 6 All-Star selections, a 2-time ERA leader, and a couple of World Series appearances.
Greinke's showing these past two seasons has left a lot to be desired. He played for the Kansas City Royals, the team that drafted him 6th overall in the 2002 MLB draft. He has pitched a total of 279.1 innings between 2022 and 2023 with a 4.38 ERA, 4.39 FIP, 1.307 WHIP, and just 5.5 strikeouts/9 innings. Greinke hasn't walked many batters, but he's been prone to allowing hits often.
If the Cardinals were to sign Greinke, it would be only temporarily unless he shows promise. If he has flashes of his excellence, then Steven Matz can move to the bullpen upon his return, a place where he saw success last year. Don't expect much out of Greinke, but at least Oli Marmol can keep Liberatore in the bullpen, a place where he, too, is seeing more success. Greinke's track record gives Marmol a bit more confidence rolling him out every 5 days as well.