How do you calm an angry customer on the telephone and have been put on hold?

The customer put YOU on hold?

I think I'd be tempted to hang up on them.
But honestly I would love to invent a thing that made these companies press 1 if you want to receive your payment this month, press 2 if you have a problem with my delinquent acct, press 3 if you want to take me to court.

I am so fed up with no being able to get hold of an actual person that I would never put them on hold.

Lower the tone in your voice - speak softly; customers will mirror your voice and will calm down. IF they start asking for a supervisor tell them that your are responsible for assitsing Customers and you can resolve their complaint; then just go into the question don't let them talk over you. Just start asking them questions, get them talking.

Can you give more details, if they put you on hold you can't say anything. What is their complaint and what business are you in?

From where I work we are allowed to reason with customers who are unreasonable (tech support) even if we get angry too, as long as we don't cross the line. If he asks for a supervisor, try to deescalate, if that doesn't work might as well give it to him, your supervisor will know anyway.

Focus on what you can do, not what you can't. If you can't give him what he wants, look for an option closest to it.

This won't always work though. There really are "Impossible Customers" sometimes.