Tink Hence
Bringing up Tink Hence, a 21-year-old who has pitched fewer than 100 innings of baseball in AA, would be the boldest of decisions. The Cardinals promoting a pitcher from Double-A has a brief history (Jordan Hicks for one), so they'll be extremely hesitant to do it this time.
Hence has two major factors playing in his favor: he lines up perfectly, and he's extremely talented. Hence last pitched on the 18th, so he'll be ready to pitch on the 23rd, which would be this Thursday. The Cardinals have an off day, so he would even get an extra day of rest, which may behoove him given his increased usage this year.
Hence is also quite talented. He's the organization's #2 ranked prospect on most lists, though Masyn Winn will graduate when the next batch comes out. Hence has a career 3.49 ERA, 1.179 WHIP, and 11 Ks/9. While some of those figures are bolstered by success at the lowest levels in baseball, his showing last year and this year has been promising.
What goes against Hence being promoted would be his age, inexperience, and a couple of less-than-ideal starts recently. He threw just 3.2 innings on May 12th, and he gave up 9 runs (none of them earned) with only 1 strikeout and 3 walks. The start prior to that Hence pitched just 4.1 innings, and he allowed 6 runs to score. He struck out 5 batters, but he walked 2 and allowed 7 hits. This was a blow-up start.
Hence has still pitched well this year in the minors; He has a 3.18 ERA, 1.084 WHIP, and he's striking out more than 10 batters per nine innings while allowing few home runs. It would be uncharacteristic and premature for the Cardinals to promote Tink Hence right now to the starting rotation, but it would be an interesting test for the organization's best pitching prospect in a long time.