This is where the volunteer EMT system of my rural hometown is superior to city EMTs.

Effectively people are on call. A radio is passed around amongst the volunteers and it's pretty open (most of my family have been EMTs at one point, and for decades).

Because it's rural, there's not a whole lot of cases needing EMTs, but they do occur. And because there's a rotating schedule and it's not anyone's full time job, it's really easy to say "not to day" or "I need a month".

I don't know how you can translate this into professional EMTs for a city. The best I got is hiring more EMTs and offering more flexible and lenient time off.