There is no 'grace' period of 1 month, it is technically 0 days after job loss or H1B expiry. However, when you enter the US, the Immigration officer at the airport may choose to give you a few days in addition to your H1B expiry date. That said, it is unlikely they will prosecute a H1B holder, if he overstays for a month or so.
IMHO the H1-B visa should have a scheme like allowing 3 months for every every year worked. Since many tech jobs are filled for 2-4 years, that would give people 6-12 months to find something new.
Out of curiosity, how much does it cost a company to takeover an existing visa?
> If a foreign worker in H-1B status quits or is dismissed from the sponsoring employer, the worker must either apply for and be granted a change of status to another non-immigrant status, find another employer (subject to application for adjustment of status and/or change of visa), or leave the US.