Data-driven models of stellar spectra are useful tools to study non-stellar information, such as the Diffuse Interstellar Bands (DIBs) caused by intervening interstellar material. Using $\sim 55000$ spectra of $\sim 17000$ red clump stars from the APOGEE DR16 dataset, we create 2nd order polynomial models of the continuum-normalized flux as a function of stellar parameters ($T_{eff}$, $\log g$, [Fe/H], [$\alpha$/Fe], and Age). The model and data show good agreement within uncertainties across the APOGEE wavelength range, although many regions reveal residuals that are not in the stellar rest-frame. We show that many of these residual features -- having average extrema at the level of $\sim3\%$ in stellar flux on average -- can be attributed to incompletely-removed spectral lines from the Earth's atmosphere and DIBs from the interstellar medium (ISM). After removing most of the remaining contamination from the Earth's sky, we identify 84 absorption features not seen in unreddened sightlights that have $<50\%$ probability of being noise artifacts -- with 25 of these features having $<5\%$ probability of being noise artifacts -- including all 10 previously-known DIBs in the APOGEE wavelength range. Because many of these features occur in the wavelength windows that APOGEE uses to measure chemical abundances, characterization and removal of this non-stellar contamination is an important step in reaching the precision required for chemical tagging experiments. Proper characterization of these features will benefit Galactic ISM science and the currently-ongoing Milky Way Mapper program of SDSS-V, which relies on the APOGEE spectrograph.

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