Zero excess in the study of geographically referenced mortality data sets has been the focus of considerable attention in the literature, with zero-inflation being the most common procedure to handle this lack of fit. Although hurdle models have also been used in disease mapping studies, their use is more rare. We show in this paper that models using particular treatments of zero excesses are often required for achieving appropriate fits in regular mortality studies since, otherwise, geographical units with low expected counts are oversmoothed. However, as also shown, an indiscriminate treatment of zero excess may be unnecessary and has a problematic implementation. In this regard, we find that naive zero-inflation and hurdle models, without an explicit modeling of the probabilities of zeroes, do not fix zero excesses problems well enough and are clearly unsatisfactory. Results sharply suggest the need for an explicit modeling of the probabilities that should vary across areal units. Unfortunately, these more flexible modeling strategies can easily lead to improper posterior distributions as we prove in several theoretical results. Those procedures have been repeatedly used in the disease mapping literature, and one should bear these issues in mind in order to propose valid models. We finally propose several valid modeling alternatives according to the results mentioned that are suitable for fitting zero excesses. We show that those proposals fix zero excesses problems and correct the mentioned oversmoothing of risks in low populated units depicting geographic patterns more suited to the data.