There is a good chance the new employer is cap-exempt but there is no way to advise without reviewing the affiliation agreement and knowing whether they are tax-exempt.
I agree with Attorney Feldman. It is highly likely the new employer is cap-exempt, but a formal assessment cannot be made without having an opportunity to review the affiliation agreement and tax exempt status.
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If the hospital is affiliated with an institution of higher education, it is a cap-exempt organization and can therefore file its H1B petition for you anytime.
The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter, nor should it be viewed as establishing an attorney client relationship of any kind.