When I left the house, Nelson's car was not around. - This means you observed that Nelson's car was not in the area when you left the house. These two things happened at the same time.
When I left the house, Nelson had already left. His car wasn't around. - This means Nelson left before you left the house, which would, of course, mean that his car was not in the area - not around - before you left the house.
The phrase "out of the house" emphasizes the idea of not being surrounded by four walls and a roof or a ceiling. Here's how you could use it:
He went out of the house without a jacket and it was freezing cold.
The sun is shining. It's a great day. You should get out of the house.
Keep Mice Out of the House Without Using Pesticides.
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