Monte Carlo algorithms such as the Wang-Landau algorithm and similar "entropic" methods are able to accurately sample the density of states of model systems and thereby give access to thermal equilibrium properties at any temperature. Thermal equilibrium is, however, unachievable at low temperatures in glassy systems. Such systems are characterized by a multitude of metastable configurations pictorially referred to as "valleys" of an energy landscape. Geometrical properties of the landscape, e.g., the local density of states describing the distribution in energy of the states belonging to a single valley, are key to understanding the dynamical properties of such systems. In this paper we combine the lid algorithm, a tool for landscape exploration previously applied to a range of models, with the Wang-Swendsen algorithm. To test this improved exploration tool, we consider a paradigmatic complex system, the Edwards-Anderson model in two and three spatial dimensions. We find a striking difference between the energy dependence of the local density of states in two dimensions and three dimensions--nearly linear in the first case, and nearly exponential in the second. The dynamical consequences of these findings are discussed.