Dispatchers' Greater Alarms were not intended to fill a FIRST ALARM assignment, it was only to supplement a SECOND or GREATER alarm ordered by the incident commander. Therefore (though I won't swear it was never done) there was no such thing as a Dispatchers' Second Alarm [to fill an incomplete first alarm assignment].
The c1961 "Manual of Fire Communications" defines it:
"1.13.2 In the event engine and ladder companies assigned on second or greater alarm are not in service, dispatcher shall provide required number by special call or greater alarm."
This was not new, it was in effect even earlier, my Regs revised to December 1945 reads in part (may have been in effect long before that date):
"When all companies assigned to respond to a 2nd or greater alarm...are not in quarters to respond, the Dispatcher shall provide the required number of special calls or a greater alarm..."
So if a multiple alarm was ordered, and insufficient companies were available, dispatchers were authorized to transmit a higher alarm, not always the next higher alarm, but even beyond that. It was a fast way to get many units on the road, rather than tapping out numerous special-calls on the bells. One bad part was that it would result in many companies proceeding on relocations per the assignment card; often relocating to companies that were not even at the fire. Last I remember this being used was mid 1970s.
There were other cases where it backfired, like some staff chiefs or special units responding where it was not warranted. One story often told was in lower Brownsville or Canarsie a unit ordered a second alarm while responding in when he saw a bright orange glow in many windows of a high rise from a distance away. Dispatchers Third or Fourth Alarm was transmitted to provide enough units for the "second". Turns out it was reflections in the high-rise windows from the setting sun.
