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Observations have shown that the disk frequency and the fraction of accreting pre-main-sequence stars decrease with the age of the population and that some stars appear to have disks while their accretion has stopped. Still, it is unclear how disk-bearing stars stop their accretion. To provide insight into the last stages of accretion in low-mass young stars, we conducted a survey of disk-bearing stars that are thought to be non-accretors to identify stars still accreting at very low rates. Here we present the first catalog of the survey of 170 disk-bearing non-accreting stars in Chamaeleon I, Orion OB1, Upper Scorpius, $γ$ Velorum, and Upper Centaurus Lupus, using He I $λ$10830 as a sensitive probe of accretion. We classify the line profiles into six types and argue that those showing redshifted and/or blueshifted absorption are still accreting. Using these classifications, we found that, among disk-bearing stars previously classified as non-accretors, at least 20-30% are still accreting, with a larger fraction of those at younger population ages. While the difference between the outer disk signature and accretion status is unclear, we find a difference between the inner disk excess and accretion status. There is no preference in the mass of the newly identified accretors, suggesting that the processes inhibiting accretion do not directly depend on mass in the typical mass range of T Tauri stars. Lastly, we found that at a low accretion level, the H$α$ width at the 10% height criteria mischaracterizes a larger fraction of accretors than the line's equivalent width.