Attracting and retaining international students has been widely discussed in higher education settings. Increasing the number of international students has become an indispensable strategy for national and global competition. This study focuses on effective strategies and international students’ issues regarding satisfaction in the most popular STEM (science, technology, engineering, and mathematics) programs. We designed a structural equation modeling (SEM) method to determine the effect of institutional mediation between push factors and satisfaction factors for the development of better strategies by which to attract and retain international students. This study employed a self-designed questionnaire to collect data: 485 degree-seeking international students in STEM programs were invited and successfully participated in this study during spring 2021 in Taiwan. IBM SPSS 26 and AMOS 26 (Analysis of Moment Structure) were used to carry out the data analysis. We employed reliability, factor, and SEM analyses. This study assumed that the impact of push factors could be modified by institutional situations and result in international students’ satisfaction with their learning and environment and regarding migration policy. The results revealed that the predictors, mediation, and criteria were significant at the 0.05 or 0.01 levels. The findings suggest that push factors impact international students’ satisfaction when using institutional leadership and strategy. The results of the bootstrap with a generalized least-squares method showed that the SEM model fit in 2000 bootstrap samples. The effect of institutional mediation can provide useful information for STEM programs to boost their future recruitment and retention strategies. This study provides an innovative approach to the detection of issues among international students in specific programs. The design of the study can be extended to similar higher education settings. These findings can enrich our knowledge regarding attracting and retaining global students in higher education.