The feedout process transfers mass perturbations from the rear to the front surface of a driven target, producing the seed for the Rayleigh-Taylor (RT) instability growth. The feedout mechanism is investigated analytically and numerically for the case of perturbation wavelength comparable to or less than the shock-compressed target thickness. The lateral mass flow in the target leads to oscillations of the initial mass non-uniformity before the reflected rippled rarefaction wave breaks out, which may result in RT bubbles produced at locations where the areal mass was initially higher. This process is determined by the evolution of hydrodynamic perturbations in the rippled rarefaction wave, which is not the same as the Richtmyer-Meshkov (RM) interfacial instability. An exact analytical formula is derived for the time-dependent mass variation in a rippled rarefaction wave, and explicit estimates are given for the time of first phase reversal and frequency of the oscillations. The limiting transition from the case of RM perturbation growth at large density difference low ambient density behind the rear surface to the case of feedout zero density is studied, and it is shown that the latter limit is approached only if the ambient density is extremely low, less than 1/1000 of the pre-shock target density.