We present a new upper limit on the energy that may be extracted from a Kerr black hole by means of particle collisions in the ergosphere (i.e., the collisional Penrose process). Earlier work on this subject has focused largely on particles with critical values of angular momentum falling into an extremal Kerr black hole from infinity and colliding just outside the horizon. While these collisions are able to reach arbitrarily high center-of-mass energies, it is very difficult for the reaction products to escape back to infinity, effectively limiting the peak efficiency of such a process to roughly $130%$. When we allow one of the initial particles to have impact parameter $b > 2M$, and thus not get captured by the horizon, it is able to collide along outgoing trajectories, greatly increasing the chance that the products can escape. For equal-mass particles annihilating to photons, we find a greatly increased peak energy of $E_{rm out} approx 6times E_{rm in}$. For Compton scattering, the efficiency can go even higher, with $E_{rm out} approx 14times E_{rm in}$, and for repeated scattering events, photons can both be produced {it and} escape to infinity with Planck-scale energies.
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