Do we need to inform USCIS when we move to old EMPLOYER if with same client?

[imiuser2504]

Hi,

I have worked with EMPLOYER-A(from APR 2015) for client XYZ at NJ location till JUNE 2016.From July 2016, i have joined EMPLOYER-B and continued working for same client XYZ at same location NJ.
I have joined EMPLOYER-B with H1B receipt number and now i got an RFE from USCIS to provide additional documents to support employer-employee relationship.
To handle this RFE situation, i have approached my previous employer EMPLOYER-A to rehire me for which he agreed. Now the question would be

1. Does EMPLOYER-A needs to file any kind of amendment to USCIS before bringing me on their payroll?

I have checked with EMPLOYER-A attorney and they confirmed that i can come on EMPLOYER-A payroll without any procedures with USCIS. As in this case, EMPLOYER-A still holds my H1B petition and there's is no change in client&location i would be working(meaning i'll be working at same client&Location as it was mentioned at EMPLOYER-As I-129 application when it was filed at APR 2015).
could someone please confirm if this is correct.

Thanks!