McGreevy was promoted to Triple-A Memphis on Saturday, Katie Woo of The Athletic reports.
McGreevy began the year with Double-A Springfield after producing an unremarkable 4.64 ERA in 20 starts there last season. The Cardinals evidently didn't need to see much to move him up to the next level, as he got the call after recording a 1.45 ERA in three starts. Getting to the doorstep of the big leagues presumably puts him in consideration for a promotion later this season, though he's not yet on the 40-man roster.