While the need for moral sensitivity amongst engineers is underlined by academic and professional standards, it is unclear whether engineering students are aware of their moral responsibilities in real-life situations or the impact they can have on social challenges. This study aims to understand the moral sensitivity of undergraduate engineering students by examining which ethical issues students are most sensitive to and the factors that are associated with increased moral sensitivity. Data were collected by conducting 52 semistructured interviews with undergraduate engineering students at two public universities. The interviews leveraged a story modified from a New York Times article about Hurricane Ida in southern Louisiana in 2021 to evaluate students' moral sensitivity. Interview data were analyzed using qualitative and quantitative methods. Of the ethical issues that students identified in the disaster case study, socioeconomic inequality was emphasized most across the interviews. We found significant relationships between moral sensitivity and the institutional contexts of students' universities, as well as the types of student organizations in which students are members. The case study of a recent disaster reveals opportunities to provide practical methods for evaluating moral sensitivity. The cultivation of engineering students' moral sensitivity through academic instruction can be promoted in a number of ways. Educators are encouraged to broaden engineering ethics education programs by supporting extracurricular activities for students in order to cultivate moral sensitivity through academic instruction.