Not solely because the first job is authorized. The additional work needs review of its eligibility and the procedure applicable to the existing status.
Plan the change before performing it
Record the proposed start date and activities, then identify the actual permission required. Travel between assignments is not an automatic method of adding an employer. Keep the first employer’s commitments visible while arranging a lawful sequence for the second proposal; neither timetable replaces the immigration requirements.
The safe sequence starts with a written confirmation that the duties fit a listed profession and the credential limb is satisfied; for a diploma-plus-experience case that includes collecting the employer letters, which can take weeks if a former employer is slow. List every dependent's step beside the party that controls it: the consular post controls appointment availability, a registry controls how quickly a certificate is issued, an inspecting officer controls admission on the day, and the employer controls the start date. The immigration step is usually the shorter one; an admission in TN status can be sought once the letter and credentials are ready, and admission is granted for a period set by the officer, with extensions possible while the employment continues.