Not enough information; the answer may require review of your documents or more specifics. If you specifically got a 17-month STEM OPT extension, then it does not cover work for an employer that is not enrolled in E-Verify, but if that employer is an accredited non-profit university then they may file a cap-exempt H-1B petition for you. For more specifics, you need to consult with an attorney.
This is general information only. It is not intended as a substitute for legal advice, and does not create an attorney-client relationship.
You should have ascertained what you needed to do before you started to work for the second employer.
The law requires that to apply for the extension, the employer must be an E-Verify Employer. There are also a requirement for certain changes (address, name, employer name, etc.) that need to be reported to the student's DSO within 10 days.
You should contact the DSO office immediately.
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While on OPT STEM you must work only for E-verified employers. If the university would like to sponsor an H-1b, then you need to speak with them.
The information you obtain in this response is for general knowledge. You should consult an attorney for advice regarding your individual situation. I invite you to contact me and welcome your calls, letters and electronic mail. Contacting me, nor my responding to the question, creates an attorney-client relationship. Please do not send any confidential information to me until such time as an attorney-client relationship has been established.
If this is an OPT extension for STEM, the employer must be e-verified.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.