We revisit the scenario of the gravitational radiation recoil acquired by the final remnant of a black-hole-binary merger by studying a set of configurations that have components of the spin both aligned with the orbital angular momentum and in the orbital plane. We perform a series of 36 new full numerical simulations for equal-mass and equal-spin-magnitude binaries, but with different spin orientations. We extend previous recoil fitting formulas to include nonlinear terms in the spins and successfully include both the new and known results. For this new formula the predicted maximum velocity approaches 5000km/s. More importantly, from the astrophysical point of view, it reaches this maximum for spins partially aligned with the orbital angular momentum. The optimal configuration is near an equipartition of the hangup and superkick contributions. This newly discovered contribution to the recoil leads to an important increase of the probabilities of large recoils in generic astrophysical mergers. We measure these probabilities for the case of accretion-aligned spins and find non-negligible probabilities for supermassive black hole encounters leading to recoil velocities of several thousand km/s.