New Killer.. Not Spirit's Father
I re-read The Spirit's lore on the wiki, and it doesn't make any sense why the new killer would be her father. She wanted revenge on her Father so the entity took her before she could die for that exact reason.. If anything, the killer is her ancestor (hence the mask and the reverse flow of time).
I could be wrong, but am I the only one who doesn't think it's Rin's father? Again, lore-wise it wouldn't make sense for the entity to do this (but then again, it's the entitiy... and it's also just a game..)