<p>The Weibull distribution is the most commonly used parametric model for analyzing censored survival data. However, it struggles to represent non-monotonic unimodal hazard functions accurately. As an alternative, the log-logistic distribution has been proposed alongside the log-normal distribution and various generalizations of the Weibull distribution to accommodate unimodal hazard shapes. The log-logistic distribution assumes that the density of the natural logarithm of lifetime follows a logistic distribution, which is symmetric about its mean. In many instances, the distribution of log-survival times is not symmetric, making the log-logistic fit potentially inadequate. In this article, we propose an extension of the log-logistic distribution that allows for skewness in the log-survival time distribution. This extension provides greater flexibility in model fitting while still accommodating unimodal and monotonic hazard shapes. We compare the nature of the hazard functions for the new model with that of the log-logistic distribution. Additionally, we applied the proposed model to compare three treatment alternatives for prolonging survival times in breast cancer patients.</p>