Does the new 60-day grace period rule for employess on an H1B visa apply to me if I quit my company voluntarily?

I am on H1-B visa working for a tech company in San Francisco bay area.

I got this email from my company’s lawyer:

60-day Nonimmigrant Grace Period
A welcome relief to many who may find themselves suddenly terminated is a 60-day grace period granted to nonimmigrant workers in the E-1, E-2, E-3, H-1B, H-1B1, L-1, O-1, and TN classifications. This means that individuals who have been terminated by their employers will now have up to 60 days to have a new employer file a petition on their behalf. (This is similar to the 60-day grace period currently afforded F-1 international students.)

I want to quit my company first, then prepare for interviews and join a different company.

Yes. But, remember once you quit your job and stay in USA on grace period, you cannot travel out of country for any emergency or any sort during that period, unless you get an approval on your H1B transfer petition of the new employer.

So it is always advisable to get a job when your are still working on the old one.