Not every three in the morning call is an emergency, and the script should find out
A valley that works nights produces overnight calls that are ordinary business, and treating every one of them as a crisis costs money in both directions.
A casino worker coming off shift at two in the morning may be calling to schedule a water heater replacement for next week. Quoted an emergency rate or dropped into a callback queue, that person books with somebody else before breakfast.
Meanwhile a genuine emergency in the same hour needs a truck immediately. One script cannot serve both, so the first two questions have to separate them: is water running right now, and can it be shut off.
Give whoever answers overnight the ability to book a normal appointment, not just to take a message. An answering service that only records names converts the least contested hour of your day into the worst performing one.
Track overnight inquiries separately. If they book at a rate anywhere near daytime calls, the staffing argument makes itself without anyone having to win a debate about it.