A new scheme to efficiently accelerate protons by a single linear polarized high-intensity ultrashort laser pulse using multiple ultrathin foils is proposed. The foils are stacked at a spacing comparable to their thickness and subsequently irradiated by the same laser pulse. The foil thicknesses are chosen such that the laser light pressure can displace all electrons out of the foil. The authors present a simple, yet precise dynamical model of the acceleration process from which both optimum foil thickness and spacing can be derived. Extensive two-dimensional (2D) particle-in-cell simulations verify the model predictions and suggest an enhancement of the maximum proton kinetic energy by 30% for the two-foil case compared to a single foil.