Holding the mast rigid will definitely help the natural frequency problem but there may be an alternative. The problem may also be prevented by wrapping your main halyard around your mast a couple of times reasonably tightly. This will disturb the flow of air around the mast so as to prevent the oscillations.

Look at a tall, slender industrial chimney sometime and you will notice that most have a spiral running around them. It is there to stop the oscillation...and they don't use stays either.

I haven't tried it myself because I've only recently bought a boat with a mast that is tall enough to worry about and I don't usually leave the mast up, however I have heard of it being used by other sailors.