We employ hydrodynamic simulations to study the effects of high shock compression ratios, as expected for fast shocks with efficient particle acceleration, on the convective instability of driven waves in supernova remnants. We find that the instability itself does not depend significantly on the compression ratio, σ, with the growth rates and the width of the mixing region at saturation being comparable for the range of ratios we studied; 4<= sigma <=21. However, because the width of the interaction region between the forward and reverse shocks can shrink significantly with increasing sigma, we find that convective instabilities can reach all the way to the forward shock front if compression ratios are high enough. Thus, if supernova blast waves accelerate particles efficiently, we expect the forward shock to be perturbed with small-amplitude, small-wavelength bumps and to find clumps and filaments of dense ejecta material in the vicinity of the shock. In addition and in contrast to situations in which sigma<=4, any enhancement of the radial magnetic field from Rayleigh-Taylor instabilities will also extend all the way to the shock front, and this may help explain the slight dominance of radial fields long seen in polarization measurements of young remnants like Tycho.