The context in Exodus 12 makes it abundantly clear that the Israelites left Egypt in the middle of the night:
At midnight the Lord struck down all the firstborn in Egypt... During the night Pharaoh summoned Moses and Aaron and said, “Up! Leave my people, you and the Israelites! Go, worship the Lord as you have requested. 32 Take your flocks and herds, as you have said, and go. And also bless me.”
33 The Egyptians urged the people to hurry and leave the country. “For otherwise,” they said, “we will all die!” 34 So the people took their dough before the yeast was added, and carried it on their shoulders in kneading troughs wrapped in clothing... The Israelites journeyed from Rameses to Sukkoth. There were about six hundred thousand men on foot, besides women and children. 38 Many other people went up with them, and also large droves of livestock, both flocks and herds. 39 With the dough the Israelites had brought from Egypt, they baked loaves of unleavened bread. The dough was without yeast because they had been driven out of Egypt and did not have time to prepare food for themselves.
For those skeptics out there, Deuteronomy 16:1 clearly spells this out:
Observe the month of Abib and celebrate the Passover of the LORD your God, because in the month of Abib he brought you out of Egypt by night.
Numbers 33:3 does not necessarily contradict that, since that could mean that they didnt actually cross the border of Egypt until dawn. But it's clear that they left Goshen when it was still dark.
But in verse 22 Moses commands them not to leave the house until morning:
Take a bunch of hyssop, dip it into the blood in the basin and put some of the blood on the top and on both sides of the doorframe. None of you shall go out of the door of your house until morning.
Did the Israelites not heed this command then? Moses knew well that the Egyptians would rush them out of Egypt, indeed Moses commands them in v. 11 to eat the Pascal lamb in haste:
This is how you are to eat it: with your cloak tucked into your belt, your sandals on your feet and your staff in your hand. Eat it in haste; it is the Lord’s Passover.
So it seems like this was part of the master plan to leave Egypt in haste in the middle of the night. So why did God command them to stay in their homes the whole night if they were meant to leave in haste in the middle of the night?? Or better yet, if they were told to stay home until morning why the need to rush through the meal and be on high alert if they weren't able to leave until dawn anyway? What gives?