Zero-inflated count data are frequently encountered in public health and epidemiology research. Two-parts model is often used to model the excessive zeros, which are a mixture of two components: a point mass at zero and a count distribution, such as a Poisson distribution. When the rate of events per unit exposure is of interest, offset is commonly used to account for the varying extent of exposure, which is essentially a predictor whose regression coefficient is fixed at one. Such an assumption of exposure effect is, however, quite restrictive for many practical problems. Further, for zero-inflated models, offset is often only included in the count component of the model. However, the probability of excessive zero component could also be affected by the amount of ``exposure''. We, therefore, proposed incorporating the varying exposure as a covariate rather than an offset term in both the probability of excessive zeros and conditional counts components of the zero-inflated model. A real example is used to illustrate the usage of the proposed methods, and simulation studies are conducted to assess the performance of the proposed methods for a broad variety of situations.