There are other ways, depending on where you live.
In Minnesota, clubs are very good about providing access to members of other clubs that host events.
If your club is hosting, say, the state amateur or any state event or USGA qualifier, usually other clubs in town will offer up tee times for free. Sometimes they might ask that you take a cart, other times not. There is a Twin Cities Senior League here where there are 18 or 20 clubs involved. Each week, the clubs not hosting give up two tee times to the club that is hosting.
Over the past few years, I've played clubs including Somerset, Interlachen, Golden Valley, Woodhill and other not so recognizable clubs because my club has hosted something.