The Teissier distribution, originally proposed by Teissier [31], was designed to model mortality due to aging in domestic animals. More recently, Krishna et al. [19] introduced the Unit Teissier (UT) distribution on the interval (0, 1) through the transformation $X=e^{-Y}$, where $Y$ follows the Teissier distribution. In their work, the authors derived several fundamental properties of the UT distribution and investigated parameter estimation using maximum likelihood, least squares, weighted least squares and Bayesian methods. Building upon this work, the present paper develops additional theoretical and inferential results for the UT distribution. In particular, closed-form expressions for single moments of order statistics and L-moments are obtained, and characterization results based on truncated moments are established. Furthermore, several alternative parameter estimation methods are considered, including maximum product of spacings, Cramér-von Mises, Anderson-Darling, right-tail Anderson-Darling, percentile and L-moment estimation, while the estimation methods previously studied by Krishna et al. [19] are also included for comparison. Extensive simulation studies under various parameter settings and sample sizes are conducted to assess and compare the performance of the estimators. Finally, the flexibility and practical utility of the UT distribution are demonstrated using a real dataset.