a) Work part-time for an employer
b) Work for multiple employers
c) Need a new visa or approval for each employer
d) Have the employer and role pass the same tests as the initial role and employer
The interesting thing is the answer to all the above questions is YES. Now that is positive in terms of the first two points as it allows people to be sponsored legally for part-time roles and thus have time to serve and work for multiple employers. Incidentally one of the employers could be a regular a full-time role and the other could be part time.
However while there is not a need for a whole new H-1B/E-3 visa stamp for each new employer if they come subsequently, they do need to be approved by the Department of Labor to get the LCA and USCIS via I-129 if done within the country for both visa types. In the case of the E-3 visa if done outside the US only the LCA is needed and in both cases if the role is done at the same time as the initial role in terms of the US Consulate interview, then both/multiple employers will actually be mentioned on the visa stamp itself in the passport. If the role(s) is subsequently approved by the USCIS then the person will receive an approved I-797 form which can serve as official proof for the right to work at that employer.
Of course it should be noted that because this official approval is required things like the Bachelor’s Degree Equivalence and Specialty Occupation conditions still apply for the E-3 Visa and while for the H-1B visa, subsequent employer approvals don’t count against the H-1B annual quota, the company, role, pay level, etc. will face the same level of scrutiny as any other role.
In terms of likelihood of approval that is a tougher one to answer as fewer people go down the legal route of getting approval for multiple employers. Most because, foreigners tend to only work for a single employer at a time, although a minority do unofficially work for multiple employers. In general if the role and pay level is seen to past all criteria as well as the company and there is no issues with you as a candidate then there is little reason the USCIS should deny an I-129 petition. Additionally this would apply seeking approval to work at more than 2 companies, however am sure there may be an unofficial limit whereby the USCIS may stop approving or even processing petitions.
If you get approval for a second employer part-time but then decide to leave your first employer and work full time at your new employer, technically speaking you should get a new LCA for the full time role being different from the part time role and possibly a new I-129 approval after that from the USCIS. Practically speaking a lot probably don’t follow this route as assume the original approval was fine but hopefully this won’t result in any issues down the line with things like Permanent Residency petitions.
In terms of when you can start working for a secondary employer, with the H-1B visa and the portability provision that applies in changing employer cases, that should allow you to officially start working upon filing your I-129 to USCIS and receiving the receipt date. Of course you would have to stop if the petitions was outright denied. As with changing employers on the E-3 visa, the start work time for a secondary employer is murky. Some may say you can just follow the same guidelines as the H-1B visa people, however that is not officially written anywhere so others may give you advice to wait until the I-129 petition is approved.