This study proposes a new flexible unit distribution derived from the generalized exponential geometric model, which is intended to model data confined to the unit interval (0, 1). The introduced distribution density shapes are extremely versatile, accommodating right-skewed, left-skewed, approximately symmetric, decreasing, U-shaped, increasing, and J-shaped. Furthermore, its hazard rate function is adaptable to bathtub, U-shaped, increasing, and J-shaped shapes, making it applicable to a wide range of real-world datasets. Essential statistical properties, such as moments, quantiles, and entropy measures, are closely derived and analyzed. Based on this distribution, a new regression model is developed to link bounded response variables to linear predictors, increasing its practical applicability. The maximum likelihood approach is used to estimate the parameters of the regression model and the suggested distribution. The performance of the maximum likelihood approach based on the suggested distribution and regression model is examined using a Monte Carlo simulation. The applicability of the regression model and the new distribution is demonstrated through real data analysis. The proposed distribution exhibits strong modeling capabilities for bounded data in the unit interval, making it highly applicable in fields such as reliability analysis, survival studies, and modeling of proportions or rates. Its superior performance over existing models, as demonstrated through simulation studies and real data applications, highlights its potential as a practical and flexible tool for applied statisticians.