One of the other needs that the Cardinals front office has identified is being able to stabilize the lineup, and preferably by adding a left-handed bat to the mix. Signing Willson Contreras, the promotion of Jordan Walker, and the continued developments their young core should help with a lot of this. If there was a clear position to add a left-handed bat though, it would be DH.
Some names that stand out in free agency this offseason include Anthony Rizzo and the aforementioned Joc Pederson. Pederson mashes right-handed pitching and would be a great platoon option at DH, while Rizzo could fill a very similar role as Pujols last season as the primary DH while spelling Goldschmidt at first base frequently.
The Cardinals interest in one of these or another DH in free agency will depend on if they trade guys like Nolan Gorman, Juan Yepez, and/or Alec Burleson. All three need to be getting consistent at-bats at the major league level, and should find a lot of those at DH. Assuming a few of those names are still around, they can hold down DH, especially if major upgrades are already made at shortstop and catcher.