Strong astrophysical shocks, diffusively accelerating cosmic rays (CRs) ought to develop CR precursors. The length of such precursor Lp is believed to be set by the ratio of the CR mean free path λ to the shock speed, i.e. Lp ∼ cλ/Vsh ∼ crg/Vsh, which is formally independent of the CR pressure Pc. However, the x-ray observations of supernova remnant shocks suggest that the precursor scale may be significantly shorter than Lp which would question the above estimate unless the magnetic field is strongly amplified and the gyroradius rg is strongly reduced over a short (unresolved) spatial scale. We argue that while the CR pressure builds up ahead of the shock, the acceleration enters into a strongly nonlinear phase in which an acoustic instability, driven by the CR-pressure gradient, dominates other instabilities (at least in the case of low β plasma). In this regime the precursor steepens into a strongly nonlinear front whose size scales with the CR pressure as Lf ∼ Lp · (Ls/Lp)2(Pc/Pg)2, where Ls is the scale of the developed acoustic turbulence and Pc/Pg is the ratio of CR to gas pressure. Since Ls ≪ Lp, the precursor scale reduction may be strong in the case of even a moderate gas heating by the CRs through the acoustic and (possibly also) the other instabilities driven by the CRs.