We present the first near all-sky yield of oscillating red giants from the prime mission data of NASA's Transiting Exoplanet Survey Satellite (TESS). We apply machine learning toward long-cadence TESS photometry from the first data release by the MIT Quick-look Pipeline to automatically detect the presence of red giant oscillations in frequency power spectra. The detected targets are conservatively vetted to produce a total of 158505 oscillating red giants, which is an order of magnitude increase over the yield from Kepler and K2 and a lower limit to the possible yield of oscillating giants across TESS's nominal mission. For each detected target, we report effective temperatures and radii derived from colors and Gaia parallaxes, as well as estimates of their frequency at maximum oscillation power. Using our measurements, we present the first near all-sky Gaia-asteroseismology mass map, which shows global structures consistent with the expected stellar populations of our Galaxy. To demonstrate the strong potential of TESS asteroseismology for Galactic archeology even with only one month of observations, we identify 354 new candidates for oscillating giants in the Galactic halo, display the vertical mass gradient of the Milky Way disk, and visualize correlations of stellar masses with kinematic phase-space substructures, velocity dispersions, and {alpha}-abundances.