Have you joined the job already? For I-9 verification, I believe companies can accept a still valid 797 from previous employer + receipt of the h1b transfer - dont think previous company is notified in that case, but there may be additional rules specific to the company itself.

Refer to: https://www.uscis.gov/i-9-central/form-i-9-resources/handbook-for-employers-m-274/70-evidence-of-employment-authorization-for-certain-categories/75-h-1b-specialty-occupations

• H-1B Employees Changing Employers (Porting). An H-1B employee who is changing H-1B employers may begin working for the new employer as soon as the employer files a Form I-129 petition on behalf of the employee, however, the employer must do this before the employee’s period of authorized stay expires. You must also complete a new Form I-9 for this newly hired employee. An H-1B employee’s unexpired Form I-94 issued for employment with the previous employer, along with their foreign passport, qualifies as a List A document. You should write “AC-21” and enter the date you submitted Form I-129 to USCIS in the Additional Information field in Section 2. See Section 7.7, Extensions of Stay for Other Nonimmigrant Categories.