In this study, we introduce a novel distribution related to the gamma distribution, referred to as the generalized incomplete gamma distribution. This new family is defined through a stochastic representation involving a linear transformation of a random variable following a distribution derived from the upper incomplete gamma function. As a result, the proposed distribution exhibits a probability density function that effectively captures data exhibiting asymmetry and both mild and high levels of kurtosis, providing greater flexibility compared to the conventional gamma distribution. We analyze the probability density function and explore fundamental properties, including moments, skewness, and kurtosis coefficients. Parameter estimation is conducted via the maximum likelihood method, and a Monte Carlo simulation study is performed to assess the asymptotic properties of the maximum likelihood estimators. To illustrate the applicability of the proposed distribution, we present two case studies involving real-world datasets related to mineral concentration and the length of odontoblasts in guinea pigs, demonstrating that the proposed distribution provides a superior fit compared to the gamma, inverse Gaussian, and slash-type distributions.