I still like my idea of there being more than the three or four symbols we commonly saw. We usually saw them on the main characters, who were high level officers aboard ship with varied responsibilities. Spock as First Officer should wear the command star but wore the sciences oval. McCoy as CMO should have medical cross but with his greater life sciences command, he wore science oval.
I'd think that much of the crew had more limited duties and responsibilities, and a more specific symbol to indicate that. Ship's historian or phaser technician or A&A officer or Records officer or shuttle pilot. Seemingly specific duties, and broadly they could fit under command/science/engineering symbols. But what the hell, I imagine there are lots of others. TMP expanded on this with some additional logos and division symbols, (communications, environmental, tactical) and many fan-based ideas have contributed as well.