Call the number of the back of the card - "Press X if you have been given a code by us". Effectively, you're calling <number on the back of the code> + <reference to queue skip>.

It would be reasonably trivial to build a phone system that lets the agent generate a OTP of sorts.

"Hey, we need to talk about your account. Call our general enquiries number on our website, press 9 and enter 'XXXXXX' to be reconnected to me."

I was just thinking about how the agent could generate ephemeral PBX extensions. OTP-like would definitely be the way to go.

Edit: perhaps the extension would be per transaction, not per-agent, and when the customer calls the extension, the agents system can automatically pull up the customer’s account. These extensions should expire, but given the length of some customer calls, and how often I’ve been disconnected from customer service lately, perhaps it should be on the order of hours, not minutes or seconds

Not a different number to call, but instead a shortcut through the usual automated phone menus - e.g. I've had a bank tell me to phone their number and then enter an extension to take me straight through to the right person.

Just ask for an extension to reach them at when you call their public number.