Binary outcomes are extremely common in biomedical research. Despite its popularity, binomial regression often fails to model this kind of data accurately due to the overdispersion problem. Many alternatives can be found in the literature, the beta-binomial (BB) regression model being one of the most popular. The additional parameter of this model enables a better fit to overdispersed data. It also exhibits an attractive interpretation in terms of the intraclass correlation coefficient. Nonetheless, in many real data applications, a single additional parameter cannot handle the entire excess of variability. In this study, we propose a new finite mixture distribution with BB components, namely, the flexible beta-binomial (FBB), which is characterized by a richer parameterization. This allows us to enhance the variance structure to account for multiple causes of overdispersion while also preserving the intraclass correlation interpretation. The novel regression model, based on the FBB distribution, exploits the flexibility and large variety of the distribution’s possible shapes (which includes bimodality and various tail behaviors). Thus, it succeeds in accounting for several (possibly concomitant) sources of overdispersion stemming from the presence of latent groups in the population, outliers, and excessive zero observations. Adopting a Bayesian approach to inference, we perform an intensive simulation study that shows the superiority of the new regression model over that of the existing ones. Its better performance is also confirmed by three applications to real datasets extensively studied in the biomedical literature, namely, bacteria data, atomic bomb radiation data, and control mice data.