I don't think you need to elaborate much. If you said to me "When John entered the hotel lobby, he was disgusted by a/the putrid smell", it would be clear that the smell was only noticeable (at that point) in the lobby. It wouldn't be clear where it was coming from but we can assume he couldn't smell it from wherever he was directly before he entered the lobby. He could have come in from the street or from a different part of the hotel.