The crisp morning air wasn't necessary for the purpose of waking her up.
As you said, you're just waking up from a nightmare.
He wasn't waking up.
His first memory was of waking up then of everything he'd done for the day.
Two straight days of vigorous outdoor activity coupled with a later-than-usual bedtime caused Dean to sleep through the six o'clock broadcast of Public Radio news, waking only when an extended arm felt an empty bed beside him.