  We analysed solar-like oscillations in 1523 $\textit{Kepler}$ red giants
which have previously been misclassified as subgiants, with predicted $\nu_{\rm
max}$ values (based on the Kepler Input Catalogue) between 280$\mu$Hz to
700$\mu$Hz. We report the discovery of 626 new oscillating red giants in our
sample, in addition to 897 oscillators that were previously characterized by
Hekker et al. (2011) from one quarter of $\textit{Kepler}$ data. Our sample
increases the known number of oscillating low-luminosity red giants by $26\%$
(up to $\sim$ 1900 stars). About three quarters of our sample are classified as
ascending red-giant-branch stars, while the remainder are red-clump stars. A
novel scheme was applied to determine $\Delta \nu$ for 108 stars with $\nu_{\rm
max}$ close to the Nyquist frequency (240$\mu$Hz < $\nu_{\rm max}$ <
320$\mu$Hz). Additionally, we identified 47 stars oscillating in the
super-Nyquist frequency regime, up to 387$\mu$Hz, using long-cadence light
curves. We show that the misclassifications are most likely due to large
uncertainties in KIC surface gravities, and do not result from the absence of
broadband colors or from different physical properties such as reddening,
spatial distribution, mass or metallicity. The sample will be valuable to study
oscillations in low-luminosity red giants and to characterize planet candidates
around those stars.
