In the Wee Small Hours is a recording by Frank Sinatra. Francis Albert Sinatra (December 12, 1915 – May 14, 1998) was an American singer who is considered one of the finest vocalists of all time, renowned for his impeccable phrasing and timing. ...
The adjectives small and wee both refer to the low numbers of those hours (one o'clock, two o'clock, etc.).
Often the start of morning is extended before dawn, this time is considered the smallhours ( weesmallhours) of the morning, so it can come to mean midnight to midday (in cultures using the 12-hour clock, these hours are ante meridiem, A.M. a.m.
It is often referred to euphemistically to refer to dawning or birth, usually with optimism, in such phrases as on a new morning or in that bright morning, for example.