We calculate the scattering amplitude of two rotating objects with the linear-in-curvature spin-induced multipoles of Kerr black holes at O(G2) and all orders in the spins of both objects. This is done including the complete set of contact terms potentially relevant to Kerr-black-hole scattering at O(G2). As such, Kerr black holes should be described by this scattering amplitude for a specific choice of values for the contact-term coefficients. The inclusion of all potential contact terms means this amplitude allows for a comprehensive search for structures emerging for certain values of the coefficients, and hence special properties that might be exhibited by Kerr-black-hole scattering. Our result can also act as a template for comparison for future computations of classical gravitational high-spin scattering.