Although the mass growth of supermassive black holes during the epoch of reionization is expected to play a role in shaping the concurrent growth of their host galaxies, observational evidence of feedback at z ≳ 6 is still sparse. We perform the first multiscale and multiphase characterization of black-hole-driven outflows in the z ∼ 6.6 quasar J0923+0402 and assess how these winds impact the cold gas reservoir. We employ the SimBAL spectral synthesis to fit broad absorption line features and find a powerful ionized outflow on a ≲210 pc scale, with a kinetic power ∼2%–100% of the quasar luminosity. Atacama Large Millimeter/submillimeter Array (ALMA) observations of [C ii] emission allow us to study the morphology and kinematics of the cold gas. We detect high-velocity [C ii] emission, likely associated with a cold neutral outflow at ∼0.5−2 kpc scale in the host galaxy, and a bright extended [C ii] halo with a size of ∼15 kpc. For the first time at such an early epoch, we accurately constrain the outflow energetics in both the ionized and the atomic neutral gas phases. We find such energetics to be consistent with expectations for an efficient feedback mechanism, and both ejective and preventative feedback modes are likely at play. The scales and energetics of the ionized and atomic outflows suggest that they might be associated with different quasar accretion episodes. The results of this work indicate that strong black hole feedback is occurring in quasars at z ≳ 6 and is likely responsible for shaping the properties of the cold gas reservoir up to circumgalactic scales.