H1B COS effective vs intended date


I am working for a company A on L1 visa and I filed my H1B through a consultant ( it was a COS) and it is approved so what I know is that I am suppose to work for the consulting company from Oct 1 and resign from my existing job with company A on L1.

But my consultant has not yet found a project for me and is asking me not to resign with company A. what he is saying is that i don’t necessarily need to join him from 1st of Oct and I can join him later once they have a project for me and they can send a note to USCIS that project got delayed becuase of some unforseen reasons thats why I am joining them late then initial intended date (i.e. 1st Oct). He is saying that I can do this upto three months ( USCIS allows that). Is it really true?

I think this is not right and he is just trying to play around as he hasn’t found a project for me.

Could you please help if this is possible else will it cause future problem with my green card processing?