In this work we follow an intermediate strategy which we may denote as quasi-direct. In this approach we construct a fitted PES, but this is done during the dynamics computation instead of before. The approach is direct in the sense that the PES is extended on the fly during the dynamics simulation with calls for new electronic structure SPs when required by the dynamics. We will utilize a PES fitted to SPs but we will only request new SPs and do refittings of the PES according to need. Compared to the approach with a pre-fitted PES, this will circumvent the unnecessary computation of SPs in domains that the wave function do not actually visit and if the molecular wave function moves towards domains not covered by the fitted PES, new SPs will be acquired so the dynamics can continue in a reliable manner. This will lead to a PES that is constructed for the specific problem and propagation of consideration. The PES is a product of the whole computation and can be reused in other computations, but the core idea is that the PES is constructed for the purpose of the dynamics computation. The quasi-direct framework is similar in spirit to some direct dynamics works on growing a PES or employing interpolation or machine learning (ML) methods such as Gaussian process regression (GPR) during the direct dynamics. In future work we return to acceleration of the present methodology with ML. In this work we wish, however, to build up a physical motivated algorithm for constructing and updating the PES during timedependent molecular quantum dynamics computations without noise and errors from ML.