This can result in two problems:
1. As you anticipated, with greencard processing ... this can be resolved by getting your greencard at a US Consul and/or exiting/re-entering the US with a new H stamp.
2. You could be deemed out-of-status when filing the new H ... and the COS can be denied for that reason.
See if you can stay with the old employer, or if the new one can file OVERNIGHT MAIL before the 22nd.
Talk to the new company' immigration lawyer.
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It is always great to stay in status. Falling out of status may result in a number of problems some of which are outlined so eloquently in my colleagues's post.
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This could be a problem. If the H-1B petition is approved, but the change of employers is not because you are found to be out of status, you will have to go abroad to obtain a new H-1B visa.
(213) 394-4554 x0 Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
It is in your new employer's best interest to make sure they file for your H-1B transfer BEFORE your first day at work (March 25) or else THEY ARE VIOLATING THE LAW by making you start on March 25. From a practical point of view, you should talk to your new employer to make sure they are confident that theire petition will be filed before that date. If they don't already have an immigration lawyer (but I think they do since you mentioned they already filed the LCA), I would tell them to consult with an attorney about requiring you to start March 25. This might be a problem not only for you but for them also.
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