We study black holes in shift-symmetric scalar Gauss–Bonnet gravity extended by a cubic Galileon interaction with a distinct energy scale. Introducing this hierarchy profoundly modifies the theory’s phenomenology. The cubic interaction allows for smaller black holes, and can generate a screening mechanism near the horizon, making large Gauss–Bonnet couplings consistent with gravitational-wave bounds. Observable quantities such as the scalar charge, the innermost stable circular orbit, and its frequency are most affected for small black holes. The resulting multi-scale effective field theory remains technically natural and offers new avenues to probe gravity in the strong-field regime.