I think many people have this small gap due to orientation, medical
screening or training purpose before becoming eligible to work for the new
employer. I think you should be fine if the gap is less than 60 days (based
on below document).
Based on the Department of labor regulations (20 CFR 655.731(c)) Even if
the H-1B nonimmigrant has not yet “entered into employment” with the
employer (as described in paragraph (c)(6)(i) of this section), the employer
that has had an LCA certified and an H-1B petition approved for the H-1B
nonimmigrant shall pay the nonimmigrant the required wage beginning 30 days
after the date the nonimmigrant first is admitted into the U.S. pursuant to
the petition, or, if the nonimmigrant is present in the United States on the
date of the approval of the petition, beginning 60 days after the date the
nonimmigrant becomes eligible to work for the employer. For purposes of this
latter requirement, the H-1B nonimmigrant is considered to be eligible to
work for the employer upon the date of need set forth on the approved H-1B
petition filed by the employer, or the date of adjustment of the
nonimmigrant's status by DHS, whichever is later. Matters such as the worker
's obtaining a State license would not be relevant to this determination.
