With the growing volume of data in society, the need for privacy protection in data analysis also rises. In particular, private selection tasks, wherein the most important information is retrieved under differential privacy are emphasized in a wide range of contexts, including machine learning and medical statistical analysis. However, existing mechanisms use global sensitivity, which may add larger amount of perturbation than is necessary. Therefore, this study proposes a novel mechanism for differentially private selection using the concept of smooth sensitivity and presents theoretical proofs of strict privacy guarantees. Simultaneously, given that the current state-of-the-art algorithm using smooth sensitivity is still of limited use, and that the theoretical analysis of the basic properties of the noise distributions are not yet rigorous, we present fundamental theorems to improve upon them. Furthermore, new theorems are proposed for efficient noise generation. Experiments demonstrate that the proposed mechanism can provide higher accuracy than the existing global sensitivity-based methods. Finally, we show key directions for further theoretical development. Overall, this study can be an important foundational work for expanding the potential of smooth sensitivity in privacy-preserving data analysis. The Python implementation of our experiments and supplemental results are available at https://github.com/ay0408/Smooth-Private-Selection.