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In a context of uncertainty quantification, the probabilistic model of a random vector at the input of a computational code is not always known. An identification of the joint distribution on a restricted sample of experimental data can lead to a bad calibration of the model. The quantity of interest estimated at the output of the code is then subject to a bi-level epistemic uncertainty that must be properly quantified. A first level arises from the statistical estimation whilst a second one comes from the identification of the probabilistic model. Each epistemic uncertainty can thus be reduced by an enrichment with new data, either by increasing the size of the estimation sample or by increasing the size of the identification sample. When gathering data is costly, it is then interesting to know which uncertainty source to reduce first, thus introducing a trade-off between simulation and physical experiment. This paper aims at presenting a sensitivity-analysis-guided enrichment procedure in a small data context to improve the estimation quality of a quantity of interest. The proposed methodology is shown to be both low cost and adaptive by introducing importance-sampling-based methods. The performance of the guided enrichment procedure is assessed on three examples. • A nested estimator allows one to consider a bi-level epistemic uncertainty source. • Enrichment in leading source answers the trade-off between simulation and physical experiment. • New data enrichment is driven by a cost-free variance based sensitivity analysis. • Multiple importance sampling allows for data reuse and auxiliary density adaptation. • The number of calls to the black box function induced by the approach is minimized. [ABSTRACT FROM AUTHOR]