Question on the 60 day grace period when changing from TN to H1B

#1
Good day,

I am on TN status sponsored by a consultant company.

The consultant company submitted my H1B application and I got selected in the lottery and on September 3 my H1B got approved. The client company ended their contract on July 31. So I am on the TN status 60 days grace period.

My consultant company and I are looking for opportunities.
Question - when the H1B becomes active on October 1st do I get another 60 days grace period?

Please kindly advise.

Thank you!
 
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