We present a new choice of initial data for binary black hole simulations that significantly improves the efficiency of high-spin simulations. We use spherical Kerr-Schild coordinates, where the horizon of a rotating black hole is spherical, for each black hole. The superposed spherical Kerr-Schild initial data reduce the runtime by a factor of 2 compared to standard superposed Kerr-Schild for an intermediate resolution spin-0.99 binary-black-hole simulation. We also explore different variations of the gauge conditions imposed during the evolution, one of which produces an additional speed-up of 1.3.