  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 42 new full numerical
simulations for equal-mass and equal-spin-magnitude binaries. We extend
previous recoil fitting formulas to include nonlinear terms in the spins and
successfully include both the new and known results. The new predicted maximum
velocity approaches 5000km/s for spins partially aligned with the orbital
angular momentum, which leads to an important increase of the probabilities of
large recoils in generic astrophysical mergers. We find non-negligible
probabilities for recoils of several thousand km/s from accretion-aligned
binaries.
