The causal impact of COVID-19 vaccine coverage on effective reproduction number R(t) under the disease control measures in the real-world scenario is understudied, making the optimal reopening strategy (e.g., when and which control measures are supposed to be conducted) during the recovery phase difficult to design. In this study, we examine the demographic heterogeneity and time variation of the vaccine effect on disease propagation based on the Bayesian structural time series analysis. Furthermore, we explore the role of non-pharmaceutical interventions (NPIs) and the entrance of the Delta variant of COVID-19 in the vaccine effect for U.S. counties. The analysis highlights several important findings: First, vaccine effects vary among the age-specific population and population densities. The vaccine effect for areas with high population density or core airport hubs is 2 times higher than for areas with low population density. Besides, areas with more older people need a high vaccine coverage to help them against the more contagious variants (e.g., the Delta variant). Second, the business restriction policy and mask requirement are more effective in preventing COVID-19 infections than other NPI measures (e.g., bar closure, gather ban, and restaurant restrictions) for areas with high population density and core airport hubs. Furthermore, the mask requirement consistently amplifies the vaccine effects against disease propagation after the presence of contagious variants. Third, areas with a high percentage of older people are suggested to postpone relaxing the restaurant restriction or gather ban since they amplify the vaccine effect against disease infections. Such empirical insights assist recovery phases of the pandemic in designing more efficient reopening strategies, vaccine prioritization, and allocation policies.