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We present the discovery and analysis of SN\,2022oqm, a Type Ic supernova (SN) detected $<1$\,day after explosion. The SN rises to a blue and short-lived (2\,days) initial peak. Early-time spectral observations of SN\,2022oqm show a hot (40,000\,K) continuum with high-ionization C and O absorption features at velocities of 4000\,km\,s$^{-1}$, while its photospheric radius expands at 20,000\,\kms, indicating a pre-existing distribution of expanding C/O material. After $\sim2.5$\,days, both the spectrum and light curves evolve into those of a typical SN Ic, with line velocities of $\sim10,000$\,km\,s$^{-1}$, in agreement with the photospheric radius evolution. The optical light curves reach a second peak at $t\approx15$\,days. By $t=60$\,days, the spectrum of \oqm\ becomes nearly nebular, displaying strong \ion{Ca}{2} and [\ion{Ca}{2}] emission with no detectable [\ion{O}{1}], marking this event as Ca-rich. The early behavior can be explained by $10^{-3}$\,\msun\ of optically thin circumstellar material (CSM) surrounding either (1) a massive compact progenitor such as a Wolf-Rayet star, (2) a massive stripped progenitor with an extended envelope, or (3) a binary system with a white dwarf. We propose that the early-time light curve is powered by both interaction of the ejecta with the optically thin CSM and shock cooling (in the massive-star scenario). The observations can be explained by CSM that is optically thick to X-ray photons, is optically thick in the lines as seen in the spectra, and is optically thin to visible-light continuum photons that come either from downscattered X-rays or from the shock-heated ejecta. Calculations show that this scenario is self-consistent.