In this paper, we show that if the minimal basically disconnected cover ${wedge}X_imath;of;X_imath$ is given by the space of fixed a $Z(X)^#$-ultrafilters on $X_imath;(imath=1,2);and;{wedge}X_1;{ imes};{wedge}X_2$ is a basically disconnected space, then ${wedge}X_1;{ imes};{wedge}X_2$ is the minimal basically disconnected cover of $X_1;{ imes};X_2$. Moreover, observing that the product space of a P-space and a countably locally weakly Lindelof basically disconnected space is basically disconnected, we show that if X is a weakly Lindelof almost P-space and Y is a countably locally weakly Lindelof space, then (${wedge}X;{ imes};{wedge}Y,;{wedge}_X;{ imes};{wedge}_Y$) is the minimal basically disconnected cover of $X;{ imes};Y$.