The honest answer about tonight books more work than the optimistic one
A caller with no heat is deciding between waiting for you and calling somebody else, and vagueness sends them to the somebody else.
When the board is full, the instinct is to soften it. We will try to get someone out today. The caller hears a maybe, hangs up, and dials the next result while your dispatcher is still typing.
A firm answer holds better. Tell them the truth: the earliest is tomorrow between eight and eleven, here is what to do tonight, and we will call if something opens. People accept a real answer far more often than an evasive one.
Give the phone team a script for the day the board is full. Include what to say about safety in general terms, and when to advise calling the gas utility or emergency services instead of waiting for a truck.
Then confirm in writing. A text with the window, the technician's name and your number reduces the cancellation that happens two hours later when a competitor calls back.