Brewer's: Small Hours of the Morning
(The). One, two, three, four, etc., before day-break. A student who sits up all night, and goes to bed at one, two, three, etc., is said to work till the small hours of the morning, or to go to bed in the small hours of the morning.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894