This is so fascinating.  You mentioned a Byrne name over the fence.....is this still a possibility? Or does it rely on there being a Hanson link back?  
So may have been an informal adoption.  Perhaps through a church connection or friend of a friend.  
Keep ahold of the story from your father as often these family stories have some truth in them but have just got a bit mixed up as they have gone down the generations,  
I wonder who lived as a neighnour on the other side?  

Do any of the names that you are DNA matched to have locations that ar the same as the ones you know of.