Falling...

When Izuku Midoriya falls into one of the old traps of the perished Boogeyman, he knows that his work is not done yet - and so does One for All. Bleeding out admists ruins, the sentient being of OFA decides to transfer it's power to the next holder but not via DNA transfer. Rather by tracing back to it's roots - to the Shigaraki bloodline. To the shock of the Vestiges, the only descendant of the bloodline does indeed exist - and OFA is passed on. But the Vestiges wake up to a site that neither of them expected - a scorched city turned to a battlefield, people dying around every corner and a 13 year old girl, the very last surviving Shigaraki fighting to live.
She chose to forget her old Japanese name, in favor of a new one - Beth Ashton.