Yes. If it's 3:00 in the afternoon, I would say:Because if you were speaking in the middle of the day, just "since midnight" would sound a bit strange.
I don't know if there's a real answer to this question -- different people might have different feelingsThank you everyone!
What's the borderline time to use it without "last night" in this case and sound idiomatic? I mean, if I say "since midnight", say, at 10 in the morning -- is it ok? Or should it be at 9... 8... or even earlier?