AFAIK (but stand to be corrected!), no departures are allowed before 0600.
I don't know if it's a ban (I haven't seen it documented anywhere), in fact the night quota restricts numbers of movements without differentiating between arrivals and departures.
But it's true that no carriers currently schedule departures before 06:00. BD used to have the occasional 05:55 departure to JED/MED, but it normally left late and rarely got airborne before 06:30.
It would make sense that those scarce pre-06:00 night quota slots are of more value to airlines for long-haul arrivals than for short-haul departures.