We use ALMA observations of CO(2-1) in 13 massive (M ∗ ≳3 1011 M ⊙) poststarburst galaxies at z ∼0.6 to constrain the molecular gas content in galaxies shortly after they quench their major star-forming episode. The poststarburst galaxies in this study are selected from the Sloan Digital Sky Survey spectroscopic samples (Data Release 14) based on their spectral shapes, as part of the Studying QUenching at Intermediate-z Galaxies: Gas, angu L→ar momentum, and Evolution ( SQuIGGL→E ) program. Early results showed that two poststarburst galaxies host large H2 reservoirs despite their low inferred star formation rates (SFRs). Here we expand this analysis to a larger statistical sample of 13 galaxies. Six of the primary targets (45%) are detected, with MH2≳3109 M ⊙. Given their high stellar masses, this mass limit corresponds to an average gas fraction of «fH2MH2/M∗ »7% or ∼ 14% using lower stellar masses estimates derived from analytic, exponentially declining star formation histories. The gas fraction correlates with the D n 4000 spectral index, suggesting that the cold gas reservoirs decrease with time since burst, as found in local K+A galaxies. Star formation histories derived from flexible stellar population synthesis modeling support this empirical finding: galaxies that quenched ≳2150 Myr prior to observation host detectable CO(2-1) emission, while older poststarburst galaxies are undetected. The large H2 reservoirs and low SFRs in the sample imply that the quenching of star formation precedes the disappearance of the cold gas reservoirs. However, within the following 100-200 Myr, the SQuIGGL→E galaxies require the additional and efficient heating or removal of cold gas to bring their low SFRs in line with standard H2 scaling relations.