Actually there are a variety of time periods:
early morning, before dawn, dawn, morning, late morning, noon (or midday), early afternoon, late afternoon, early evening, sunset, evening, late evening, dusk, twilight, night, late at night, past midnight, in the middle of the night
plus any number of variations, colloquialism, and poetic expressions for each of these like:
lunchtime, bedtime, suppertime, the wee hours of the morning, when the cows come home, etc.
The term to use for the time from midnight until dawn depends on what you are doing. "Night" is sufficient if it is dark, but "early morning" can work if you are awake. Some other examples:
Expressions like "the predawn hours" depend on when the sun rises for that location and time of year, which in some cases can be very early.
"Past (my) bedtime" is useful to express you are awake when you should be asleep, and can imply past midnight depending on the context.
"In the dead/dark/still of (the) night" is a poetic expression that means "when it is very dark and most people are normally asleep, and everything is very quiet" -- which again can mean various times depending on location.
The witching hour is a literary expression that means "exactly midnight", supposedly when witches come out to do what witches will do.