[en] We present simulation results of the build-up of the electron-cloud density n_e for the two proposed ILC damping ring lattices, DC04 and DSB3, with particular attention to the potential benefit of an antechamber. We examine a field-free region and a dipole bending magnet, with or without an antechamber. We assume a secondary electronemission model for the chamber surface based on approximate fits to measured data for TiN, except that we let the peak value of the secondary emission yield (SEY), (delta)_max, be a variable. We conclude that there is a critical value of (delta)_max below which the antechamber provides a substantial benefit, roughly a factor ∼40 reduction in n_e relative to the case in which max exceeds the critical value. We estimate the steady-state value of n_e as a function of (delta)_max, and thereby obtain the critical value of (delta)_max for all cases considered. Thus, from the perspective of the electron-cloud effect, the inclusion of an antechamber in the design is justified only if (delta)_max is below the critical value. The results presented here constitute a slight extension of those previously presented in March and September, 2010.