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Readmission following discharge from an initial hospitalization is a key marker of quality of health care in the United States. For the most part, readmission has been used to study quality of care for patients with acute health conditions, such as pneumonia and heart failure, with analyses typically based on a logistic-Normal generalized linear mixed model. Applying this model to the study readmission among patients with increasingly prevalent advanced health conditions such as pancreatic cancer is problematic, however, because it ignores death as a competing risk. A more appropriate analysis is to imbed such studies within the semi-competing risks framework. To our knowledge, however, no comprehensive statistical methods have been developed for cluster-correlated semi-competing risks data. In this paper we propose a novel hierarchical modeling framework for the analysis of cluster-correlated semi-competing risks data. The framework permits parametric or non-parametric specifications for a range of model components, including baseline hazard functions and distributions for key random effects, giving analysts substantial flexibility as they consider their own analyses. Estimation and inference is performed within the Bayesian paradigm since it facilitates the straightforward characterization of (posterior) uncertainty for all model parameters including hospital-specific random effects. The proposed framework is used to study the risk of readmission among 5,298 Medicare beneficiaries diagnosed with pancreatic cancer at 112 hospitals in the six New England states between 2000-2009, specifically to investigate the role of patient-level risk factors and to characterize variation in risk across hospitals that is not explained by differences in patient case-mix.