You are cap exempt and need not be counted again, so technically, you can change to H-1B at any time. Having said that, however, you need to make sure that you and the position qualify for H-1B status, so consult with an immigration attorney to prepare the H-1B petition.
You are CAP except.
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You sound like you value your presence in the United States by describing all actions on your COS. To complete this process legally competently you need a good attorney not an assemblage of random legal counsels from all over the nation who never seen one legal document from your case.
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I agree with my colleagues.
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I agree with my colleagues. If the employer doesn't have an attorney assisting them on this, it might be a good idea.
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