Fair reachability is a very useful technique in detecting errors of deadlocks and unspecified receptions in networks of communicating finite state machines (CFSMs) consisting of two machines. The paper extends the classical fair reachability technique, which is only applicable to the class of two-machine CFSMs, to the general class of CFSMs. For bounded CFSMs, the extended fair reachability technique reduces by more than one half the total number of reachable global states that have to be searched in verifying freedom from deadlocks. The usefulness of the new reachability technique, called even reachability, is demonstrated through two examples.