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We explore how the presence of detectable molecular gas depends on the inferred star formation histories (SFHs) in 8 massive, quiescent galaxies at $\mathrm{z\sim0.7}$. Half of the sample have clear detections of molecular gas, traced by CO(2-1). We find that the molecular gas content is unrelated to the rate of star formation decline prior to the most recent 1 Gyr, suggesting that the gas reservoirs are not leftover from their primary star formation epoch. However, the recent SFHs of CO-detected galaxies demonstrate evidence for secondary bursts of star formation in their last Gyr. The fraction of stellar mass formed in these secondary bursts ranges from $\mathrm{f_{burst}\approx0.3-6\%}$, and ended between $\mathrm{t_{end\mbox{-}burst}\approx0-330~Myr}$ ago. The CO-detected galaxies form a higher fraction of mass in the last Gyr ($\mathrm{f_{M_{1Gyr}}=2.6\pm1.8\%}$) compared to the CO-undetected galaxies ($\mathrm{f_{M_{1Gyr}}=0.2\pm0.1\%}$). The galaxies with gas reservoirs have enhanced late-time star formation, highlighting this as a contributing factor to the observed heterogeneity in the gas reservoirs in high-redshift quiescent galaxies. We find that the amount of gas and star formation driven by these secondary bursts are inconsistent with that expected from dry minor mergers, and instead are likely driven by recently-accreted gas i.e., gas-rich minor mergers. This conclusion would not have been made based on $\mathrm{SFR_{UV+IR}}$ measurements alone, highlighting the power of detailed SFH modeling in the interpretation of gas reservoirs. Larger samples are needed to understand the frequency of low-level rejuvenation among quiescent galaxies at intermediate redshifts, and to what extent this drives the diversity of molecular gas reservoirs.