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QUESTION: If am let go by my current employer, I would like to know very specifically how long can I stay in the US while trying to find new employment, and what my various options would be?
ANSWER: There is no "grace period" once you've lost your job. The day after you are terminated, the day after you fail to maintain status, or the day after the expiration of your I-94 is the first day that you are out of status and unlawfully present in the US.
You cannot stay in the US looking while trying to find new employment. Why? Because once you fall out of status, you have no status from which to change or extend your status (which is technically the terms used for new employment on the same status).
You may have heard of a 10-day "grace period." That does not exist for E-3 status. Immigration authorities generally overlook a "reasonable" amount of time after someone's status ends, to give them time to tie up their affairs and depart the US. Usually, 10 days is that reasonable amount of time. But it is not a "grace period."
QUESTION: What my various options would be?
ANSWER: The only safe way to do this is to apply for a change of status or change of employer before your current employment is terminated. If you do not have any prospects for alternative E-3 employment immediately, and you know you will be terminated soon, you might consider applying for a change of status to B2 visitor status (which can be approved for up to 6 months) and work out your next steps during that time.
If you try to do that after you have already been terminated, there is a high likilihood that your change of statu will be denied, for failure to show that you maintained lawful status in the US throughout your stay.
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