We introduce a new technique called ‘Measure-Rewind- Measure’ (MRM) to achieve tighter security proofs in the quantum random oracle model (QROM). We first apply our MRM technique to derive a new security proof for a variant of the ‘double-sided’ quantum One- Way to Hiding Lemma (O2H) of Bindel et al. [TCC 2019] which, for the first time, avoids the square-root advantage loss in the security proof. In particular, it bypasses a previous ‘impossibility result’ of Jiang, Zhang and Ma [IACR eprint 2019]. We then apply our new O2H Lemma to give a new tighter security proof for the Fujisaki-Okamoto (FO) transform for constructing a strong (IND-CCA) Key Encapsulation Mechanism (KEM) from a weak (IND-CPA) public-key encryption scheme satisfying a mild injectivity assumption.