Jake Woodford had a disappointing 2023 season. After earning a rotation spot to start the season while Adam Wainwright was on the IL, Woodford never found his footing, and finished the year appearing in just 15 games for St. Louis with a 6.23 ERA, and finished the year in the Flordia Complex League.
If Woodford was still a pre-arbitration player, I'd feel pretty confident that he'd get a shot to figure things out this offseason and come back ready to prove himself in Spring Training. But this offseason, he'll be hitting arbitration, and although that number won't be super high, it'll be significant enough that I think the Cardinals would rather rely on their internal depth options than keep Woodford.
Zack Thompson, Matthew Liberatore, Drew Rom, Gordon Graceffo, Michael McGreevy, Adam Kloffenstein, and Sem Robberse all have stronger arguments to be the "next man up" over Woodford in the rotatoin. Unless the Cardinals see a way to transition him into a full-time reliever this offseason, I think he'll be non-tendered here soon.