Household is an important venue of transmission of infectious diseases, and it is crucial to estimate transmissibility of the pathogens, associated risk factors and effectiveness of interventions in households as well as other close contact settings. However, many key quantities for analyzing transmission dynamics are missing by nature, e.g., source and time of infection, making the likelihood difficult to track or integrate. These quantities are highly dependent on each other and traditional numerical techniques often encounter convergence problems. Based on a Bayesian continuous-time transmission model for clusters, we showed that summing over potential infectors of each infected person to base inference on the marginal likelihood does not necessarily improve estimation or mixing, as the underlying range of potential infection time becomes much wider, in comparison to sampling infection time conditioning on a sampled infector. We devised an efficient sampling scheme where infection times are sampled by households with an adaptive proposal that takes into account updated risk profiles for each individual.