Our particular on-call agreement is not 24/7 but has some periods where nobody is on call.
At the moment, when a caller calls the on-call number, one of three things will happen:
If an operator is on call, and they pick up the phone, then the caller speaks with the operator.
If an operator is on call, and they do not pick up the phone in time, then the caller leaves a message (which is transcribed and made into an incident, and the audio is available on twilio).
If nobody is on call, the phone rings indefinitely until the caller hangs up in frustration.
We would prefer that, if nobody is on call, the caller leaves a message, as in the second case.
Our setup is pretty simple, with just one rotation every day from 8:00 a.m. through midnight and with a handoff every week. The escalation policy has one step: notify the on-duty user(s) in rotation.
The only way I can think to fix this would be to add a bot user with some phone number that will never pick up that assumes on-call from midnight to 8 a.m., but (a) that seems pretty hacky, and (b) it doesn't appear possible to add non-human users anyway.