  Quenched post-starburst galaxies (QPSBs) are a rare but important class of
galaxies that show signs of rapid cessation or recent rejuvenation of star
formation. A recent observation shows that about half of QPSBs have large
amounts of cold gas. This molecular CO sample is, however, too small and is not
without limitations. Our work aims to verify previous results by applying a new
method to study a uniformly selected sample, more than 10 times larger. In
particular, we present detailed analysis of H$\alpha$/H$\beta$ ratios of
face-on QPSBs at $z = 0.02 - 0.15$ and with $M_\star =
10^{10}-10^{11}\,M_\odot$. We interpret the H$\alpha$/H$\beta$ ratios by
applying our recent gas mass calibration, which is based on non-PSB galaxies
but predicts gas masses that are consistent with CO observations of $\sim 100$
PSBs. We estimate the molecular gas by either using PSBs with well-measured
H$\alpha$/H$\beta$ ratios or by measuring them from stacked spectra. Our
analysis reveals that QPSBs have a wide range of H$\alpha$/H$\beta$ ratios and
molecular gas fractions that overlap with the typical gas fractions of
star-forming or quiescent galaxies: H$\alpha$/H$\beta \approx 3-8$ and
$f_\mathrm{H_2} \approx 1\%-20\%$ with median $f_\mathrm{H_2} \approx 4\%-6\%$,
which correspond to $M_\mathrm{H_2} \approx (1-3) \times 10^{9} \,M_\odot$. Our
results indicate that large reservoirs of cold gas are still present in
significant numbers of QPSBs, and that they arguably were not removed or
destroyed by feedback from active galactic nuclei.
