Circular distributions play a crucial role in modeling data characterized by angular properties, offering indispensable tools for analyzing angles, phases, or periodic events. The versatility of these distributions is evident in their application across various domains. There are various strategies available for constructing circular distributions. The exponential distribution is one of the most important models for analyzing lifetime data. In this work, we discuss the wrapped exponential distribution and its properties. Furthermore, we propose three extensions to the wrapped exponential distribution based on the Marshall-Olkin, type I half logistic, and exponentiated generalized generators. We present several mathematical characteristics of these extensions and a unique linear representation of their densities. We investigate the maximum likelihood, least squares, and weighted least squares estimators of the unknown parameters and conduct a simulation study to evaluate their performance. Finally, we compare our novel models against the wrapped exponential and transmuted wrapped exponential distribution using real data in four applications.