For evaluated nuclear cross-section uncertainties, most standard approaches are based on experimental cross-section measurements, reflecting that these measurements have uncertainty on their own and, in particular, undetermined correlations. We propose here focusing on the estimation of experimental covariances and bypassing the direct empirical estimator, which cannot be used due to the small amount of available data. Because of the nonlinearity of experimental cross sections, an alternative method to the classical propagation error formula is presented. This method exploits a regression model of the experimental cross sections to generate pseudomeasurements and thereby allows an empirical estimation of experimental covariances. Moreover, thanks to a bootstrap, a quality measure for the estimation is provided. The empirical matrix estimation is then improved with shrinkage. The validity of the approach is confirmed through numerical experiments on a toy model. Finally, the procedure is applied to the real case of the 5525Mn nucleus.