  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, $\gamma$ Velorum, and Upper Centaurus Lupus, using He I
$\lambda$10830 as a sensitive probe of accretion. We classify the line profiles
into six types and argue that those showing redshifted and/or blueshifted
absorption are still accreting. Using these classifications, we found that,
among disk-bearing stars previously classified as non-accretors, at least
20-30% are still accreting, with a larger fraction of those at younger
population ages. While the difference between the outer disk signature and
accretion status is unclear, we find a difference between the inner disk excess
and accretion status. There is no preference in the mass of the newly
identified accretors, suggesting that the processes inhibiting accretion do not
directly depend on mass in the typical mass range of T Tauri stars. Lastly, we
found that at a low accretion level, the H$\alpha$ width at the 10% height
criteria mischaracterizes a larger fraction of accretors than the line's
equivalent width.
