In the Dagan & Tulin (J. Fluid Mech., vol. 51, 1972, pp. 529–543) model of ship waves, a blunt ship moving at low speeds can be modelled as a two-dimensional semi-infinite body. A central question for these reduced models is whether a particular ship design can minimize, or indeed eliminate, the wave resistance. In the previous part of our work (Trinh et al., J. Fluid Mech., vol. 685, 2011, pp. 413–439), we demonstrated why a single corner can never be made waveless. In this accompanying paper, we continue our investigations with the study of more general piecewise-linear, or multi-cornered ships. By using exponential asymptotics, we demonstrate how the production of waves can be directly ascertained by the positions and angles of the corners. In particular, this theory answers the question raised by Farrow & Tuck (J. Austral. Math. Soc. B, vol. 36, 1995, pp. 424–437) as to why certain bulbous-like obstructions can minimize the production of waves. General results for wavelessness are given for a class of hulls, and numerical computations of the nonlinear ship-wave problem are used to confirm analytical predictions. Finally, we discuss open questions regarding hulls without corners and more general three-dimensional bluff bodies.