We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a random field, typically assumed Gaussian. In this setting, the posterior distribution on the objective function gives rise to a poste-rior distribution of excursion sets. Several approaches exist to summarize the distribution of the excursion sets based on random closed set the-ory. While the recently proposed Vorob'ev approach leads to analytically tractable expectations, further notions of variability require Monte Carlo estimators relying on Gaussian random field conditional simulations. In the present work we propose a method to choose simulation points and obtain realizations of the conditional field at fine designs through affine predictors. The points are chosen optimally in the sense that they mini-mize the expected distance in measure between the posterior excursion set and its reconstruction. The proposed method reduces the computational costs due to simulations and enables the prediction of realizations on fine designs even in large dimensions. We apply this reconstruction approach to obtain realizations of an excursion set on a fine grid which allow us to give a new measure of uncertainty based on the distance transform of the excursion set. Finally we present an application of the method where the distribution of the volume of excursion is estimated in a six-dimensional example.