Summary: In a previous paper of the author [Phys. Rev. D 78, 085003, 5 pages, (2008); arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as \(\mathcal H^{\text{(auxiliary)}}\) and \(\mathcal H^{\text{(standard)}}\) we spot a weak point of the 2HS formalism which lies in the double role played by \(\mathcal H^{\text{(auxiliary)}}\). As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac’s bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator \(H_{\text{(gen)}}\) of the time-evolution of the wave functions may differ from their Hamiltonian \(H\).