For statistical inference after model selection with the same data, the classical theory of statistical inference may not be valid, known as the double-dipping problem. Zrnic and Jordan (2023) proposed an algorithmic stability method that incorporates randomization to enable post-selection corrections without computational burdens. However, it suffers from the conservativeness of the confidence intervals for algorithms with complex composition structures due to the inequality-based definition of stability. To this end, we propose a new Gaussian Algorithmic Stability (GAS) framework, introducing a hypothesis-testing-based f-stability concept that redefines algorithmic stability. Theoretically, we establish the coverage guarantee of the proposed GAS method and also examine the regimes under which it yields narrower confidence intervals. In practice, the GAS is easier to tune the stability parameter by leveraging Gaussian noise than the existing method. In addition, we propose a new stability amplification approach to characterize how stability improves under contractive iterative algorithms, enabling tighter stability parameters and consequently narrower confidence intervals. Simulations and a real-data study confirm the validity and effectiveness of the proposed method.