The effects of laser-pulse polarization on the generation of an electrostatic shock in an overdense plasma were investigated using particle-in-cell simulations. We found, from one-dimensional simulations, that total and average energies of reflected ions from a circular polarization- (CP) driven shock front are a few times higher than those from a linear polarization- (LP) driven one for a given pulse energy. Moreover, it was discovered that the pulse transmittance is the single dominant factor for determining the CP-shock formation, while the LP shock is affected by the plasma scale length as well as the transmittance. In two-dimensional simulations, it is observed that the transverse instability, such as Weibel-like instability, can be suppressed more efficiently by CP pulses.