Your husband can't work part-time for a second employer under his current H-1B. He has two choices: he can volunteer or he can ask the second employer to sponsor him for a part-time H-1B visa to cover this employment. This assumes that the part-time job would meet the requirements for an H-1B visa namely a position that requires a baccalaureate. I would suggest your husband confer with an immigration attorney.
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Your husband can get a concurrent H-1B visa for part time work. I have processed part time H-1B visas. I would suggest your husband confer with an immigration attorney. My offices are in Sunnyvale.
You would need an H-1b through that school. H-1B only authorizes employment with the petitioning employer.
You should retain an experienced immigration lawyer to review all the facts and advise you accordingly.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship.