May an F-1 student with a STEM OPT extension work part-time for a university that only uses E-Verify for employees assigned to a federal contract if the student was verified through another E-Verify employer that creates cases for all new hires?

No, every employer who wishes to employ F-1 students with STEM OPT extensions must be enrolled in and in good standing with E-Verify. Each employer must enroll the hiring site or location where the F-1 STEM OPT employee will work in E-Verify, and verify all new hires at each site, including the F-1 STEM OPT student, if he or she is a new hire.