A three-dimensional object printer varies the finish in exposed surfaces of a printed object. The printer includes a controller operatively connected to at least two ejector heads, a leveling device, a curing device, and an actuator that is operatively connected to a member to which the at least two ejector heads, the leveling device, and the curing device are mounted. The controller is configured to detect a next layer to be formed being within a predetermined number of layers before an exposed surface is formed, to modify rendered data for the layers within the predetermined number of layers received from the source of rendered data, and to operate a plurality of ejectors in the at least two ejector heads with reference to the modified rendered data to apply a finish to the exposed surface.