The full phrase is 'free, gratis, and for nothing'. This makes use of the 'Rule of Three' in rhetoric, where something is said three times in one form or another. This adds emphasis, and has been found historically to be better remembered by those who hear it. 'Wee small hours of the morning' has similar intent.
Rule of Three...
The earliest recorded use of the phrase, "the wee sma' hours" was in Charlotte Bronte's 'Shirley', published in 1849. Just as a matter of interest, this predates Sinatra' s birth, never mind his song, by 66 years, Boxy!