H1B 60 days grace period extension

My employer ended my employment due to COVID reasons on May 31, 2020 but I never received an official termination letter. Calculating the grace period depending on the assumed termination date mentioned above, I have till July 30 to find a new job. Should my new employer give me an offer or start the H1B transfer application before July 30? What happens if they fail to do so in time? Can I extend the grace period on the basis of having an offer in hand? My final interviews with potential employers are being withdrawn because of the time restriction.

There is no extension available for H1B 60 day grace period.