In order to estimate the unknown size of the population that is difficult or hidden to enumerate, the capture-recapture method is widely used for this purpose. We propose the one-inflated, zero-truncated geometric (OIZTG) model to deal with three important characteristics of some capture–recapture data: zero-truncation, one-inflation, and observed heterogeneity. The OIZTG model is generated by two distinct processes, one from a zero-truncated geometric (ZTG) process, and the other one-count producing process. To explain heterogeneity at an individual level, the OIZTG model provides a simple way to link the covariate information. The new estimator was proposed based on the OIZTG distributions through the modified Horvitz-Thomson approach, and the parameters of the OIZTG distributions are estimated by using a maximum likelihood estimator (MLE). With regard to making inferences about the unknown size of the population, confidence interval estimations are proposed where variance estimate of population size estimator is achieved by using conditional expectation technique. All of these are assessed through simulation studies. The real data sets are provided for understanding the methodologies.