How to extend a 2x6 joist to accommodate a hanger board for stairs?
I am going to assume the opening for the stairs is on the far side of the picture.  

I'd skip the ties and use 2x12 hangers on the posts.  If you have this area open you might want to tie the 2x6 joists to the 4x4 posts using ties for seismic/wind though.

Why does the current board not work?