The statistical Finite Element Method (statFEM) has demonstrated significant potential for inferring displacement and strain fields from sparse and noisy data, as shown in works by (Girolami, et al., 2021) and (Febrianto, et al., 2022). However, statFEM fails to reliably infer hidden quantities, i.e. the stress field. This is especially true, if the prior differs too heavily from the observational data, see (Narouie, et al., 2023). In an offline stage, a stochastic prior displacement field is computed using Polynomial Chaos Expansion (PCE), followed by an online Bayesian update of this prior in light of observational data. This posterior is then used to compute the push-forward stress, which serves as an initial estimate of the stress field. Recognizing that this stress field is incorrect see (Narouie, et al., 2023), we introduce an ANN-based stress corrector and embed it into the weakly formulated finite element residual. Minimizing this residual, a corrected stress-field is obtained as a result of satisfying the balance of linear and angular momentum. With the extracted stress field and the associated strain data, the next step is to identify the right constitutive model. This can be achieved with the help of highly parametric interpretable material models and LASSO regression as proposed in the context of EUCLID (Moritz, et al., 2023). Overall, this consistutes a staggered approach for hidden stress field inference from uncertain observational data. We demonstrate its efficacy by means of numerical examples.