A method of operating a three-dimensional object printer has been developed. The method receives image data corresponding to at least one three-dimensional object. The method generates image data corresponding to at least one member that extends from the at least one three-dimensional object corresponding to the received image data. The method operates an ejector head of the three-dimensional object printer with a controller referencing the received image data and the generated image data to eject drops of a build material onto a platen to form the at least one three-dimensional object and the at least one member extending from the at least one three-dimensional object.