In this paper, we extend the work of Pimentel et al. (2015) and propose an adjusted estimator of Kendall's $τ$ for bivariate zero-inflated count data. We provide achievable lower and upper bounds of our proposed estimator and show its relationship with current literature. In addition, we also suggest an estimator of the achievable bounds, thereby helping practitioners interpret the results while working with real data. The performance of the proposed estimator for Kendall's $τ$ is unbiased with smaller mean squared errors compared to the unadjusted estimator of Pimentel et al. (2015). Our results also show that the bound estimator can be used when knowledge of the marginal distributions is lacking.