Can I work as a part-time employee if I have a H-1B visa from another company?
I would still be working as a full-time in the sponsor company.
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No. While on H-1b status, you can only work for the employer that petitioned you for H-1b status. You are authorized to work only through 8 CFR 274a.12(b)(9):
(b) Aliens authorized for employment with a specific employer incident to status or parole. The following classes of aliens are authorized to be employed in the United States by the specific employer and subject to any restrictions described in the section(s) of this chapter indicated as a condition of their parole or of their admission in, or subsequent change to, the designated nonimmigrant classification. An alien in one of these classes is not issued an employment authorization document by DHS:
(9) A temporary worker or trainee (H-1, H-2A, H-2B, or H-3), pursuant to 8 CFR 214.2(h), or a nonimmigrant specialty occupation worker pursuant to sections 101(a)(15)(H)(i)(b)(1), 101(a)(15)(H)(ii)(a), 101(a)(15)(H)(ii)(b) and INA 101(a)(15)(H)(iii) of the Act. An alien in this status may be employed only by the petitioner through whom the status was obtained. [...]
Furthermore, an employer other than the petitioning employer, that is following the rules, should not hire you, since you do not have the documents to pass the I-9 process. For your own employer, you can use your passport plus your H-1b I-94 as a List A document for the I-9, since the employer got an approval for the H-1b petition they filed for you, but that will not work for any other employer who did not petition you for H-1b.