The sources of ultra-high-energy cosmic rays are still unknown, but assuming standard physics, they are expected to lie within a few hundred megaparsecs from us. Indeed, over cosmological distances cosmic rays lose energy to interactions with background photons, at a rate depending on their mass number and energy and properties of photonuclear interactions and photon backgrounds. The universe is not homogeneous at such scales, hence the distribution of the arrival directions of cosmic rays is expected to reflect the inhomogeneities in the distribution of galaxies; the shorter the energy loss lengths, the stronger the expected anisotropies. Galactic and intergalactic magnetic fields can blur and distort the picture, but the magnitudes of the largest-scale anisotropies, namely the dipole and quadrupole moments, are the most robust to their effects. Measuring them with no bias regardless of any higher-order multipoles is not possible except with full-sky coverage. In this work, we achieve this in three energy ranges (approximately $8$--$16~EeV$, $16$--$32~EeV$, and $32$--$\infty~EeV$) by combining surface-detector data collected at the Pierre Auger Observatory until 2020 and at the Telescope Array (TA) until 2019, before the completion of the upgrades of the arrays with new scintillator detectors. We find that the full-sky coverage achieved by combining Auger and TA data reduces the uncertainties on the north-south components of the dipole and quadrupole in half compared to Auger-only results.