We report that a secondary, Rayleigh‐Taylor type instability can exist in cosmic‐ray dominated media which are perturbed by the 1D acoustic instability. Using the local WKB approximation, the growth rate of the secondary instability is shown to be comparable to that of the 1D acoustic instability itself in the cases we have considered. We show that flows in the precursor and postshock regions can become highly turbulent due to the secondary instability. As in the 1D acoustic instability, however, the cosmic‐ray pressure is not significantly affected by the presence of the secondary instability, because the cosmic ray diffusion timescale from the perturbations is much smaller than the growth timescale of the instability.