My H1 starts from Oct 1, recently the name of my company( same company) changed slightly.

Usually the petitioner, or the employer company files for a H-1B amendment for each of  its H or L employees.

However, USCIS’ Aleinikoff policy memorandum (8-22-96) regarding amendment of
H-1B petitions states that “changes in the ownership structure of the
petitioning entity generally do not require the filing of a new or
amended petition if the petitioning entity continues to remain the
alien beneficiary’s employer, provided the new owner(s) of the firm
assumes the previous owner’s duties and liabilities...” This has been
interpreted by USCIS Adjudications to apply to acquisition of discrete
functional components or divisions of companies.

An amendment (10/30/00) to INA §214 (c) provides that “[a]n amended
H-1B petition shall not be required where the petitioning employer is
involved in a corporate restructuring, including but not limited to a
merger, acquisition, or consolidation, where a new corporate entity
succeeds to the interests and obligations of the original petitioning
employer and where the terms and conditions of employment remain the
same but for the identity of the petitioner.”