I live in an antique home with 2x6 ceiling joists. The stairs are being rebuilt and the current hanger board for the stringers will not work. So I have attached a picture of the current setup and one of my idea to extend the joist down using a 2x12 and strong ties securing it to two adjacent 4x4 posts. Please let me know your thoughts on the strength of this design. Or any improvements or different ways to extend it down.
There is only about 2 inches behind the joist and posts to accommodate anything do to a finished room and basement stairway. This is why I've taken a minimalist approach.