We aim to further study the global stability of Boolean control networks (BCNs) under aperiodic sampleddata control (ASDC). According to our previous work, it is known that a BCN under ASDC can be transformed into a switched Boolean network (SBN), and further global stability of the BCN under ASDC can be obtained by studying the global stability of the transformed SBN. Unfortunately, since the major idea of our previous work is to use stable subsystems to offset the state divergence caused by unstable subsystems, the SBN considered has at least one stable subsystem. The central thought in this paper is that switching behavior also has good stabilization; i.e., the SBN can also be stable with appropriate switching laws designed, even if all subsystems are unstable. This is completely different from that in our previous work. Specifically, for this case, the dwell time (DT) should be limited within a pair of upper and lower bounds. By means of the discretized Lyapunov function and DT, a sufficient condition for global stability is obtained. Finally, the above results are demonstrated by a biological example.