Sully did go off script. It's because he knew the aircraft very well. For the record I have flown with him on several occasions. Starting the APU is on the second page of the Engine Out portion of the QRH. That dual flameout provision was / is for high altitude flame out. Sully was old school Airforce and had a lot committed to memory, even if it wasn't required.

The important thing about starting the APU out of sequence was not necessarily getting an airstart. It was keeping the computers on line and staying in Normal Law.

If you have seen the Tom Hanks movie, it is accurate to a T because they used flight and cockpit voice recorder data. The simulator scenes were flown by US Airways check-airmen.

In one scene in the cabin, the lights went out. That was the generators dropping off line. The engines were windmilling enough to give erratic power and hydraulics. Starting the APU guaranteed good computers and Alpha protection. Erratic power and hydraulics could have led to an Alternate Law situation where you lose Alpha Floor protection. With the APU on it also guaranteed blue hydraulics at full power, which supports normal flight controls.

Alpha Floor protection enabled Sully to hold full aft stick prior to impact and it allowed the aircraft to settle into the water without stalling. There is not a great deal of technique required, just keep the wings level and the aircraft will land at the slowest possible speed with a nose up attitude.

The movie also stated that the engines were producing power. What they were looking at was the auto start function. When the engines registered a failure on ECAM, placing the start switch in air start turned on ignition and introduces start fuel. The engines would accelerate however due to the damage the egt limits were hit and the sequence would start over again. The engines were so badly damaged they wouldn't run without over-temping.