From Murthy.com
If one's H-1 is expiring after 6 yrs, can he get 60 days grace period to stay in the country and then leave?
Attorney Murthy : There is no grace period of 60 days after completion of the 6-year term on H1B. Only an F-1 student, after completion of the degree program, enjoys a 60-day grace period. All others have different grace periods. A person on an H1B status has 10 days before the start of the H1B and 10 days at the end of the H1B status to stay and wrap up matters, pack, and depart the U.S. If one chooses to remain for 60 days, s/he violates the terms of the H1B and then all future visa applications will need to be applied for in the home country. Also, there is a slight risk of being removable or deportable when one overstays one's legal status even by a single day. A possible option for one who needs more time could be to file a COS to B-1/B-2 to wrap up matters, though it is likely that the USCIS will not make a decision on the B-1/B-2 until after the person departs the U.S. There is a benefit in filing the COS since then there is no unlawful presence, and one should obtain a copy of the approval notice to show the consulate down the road if the matter is investigated or if s/he ever needs a visa to the U.S. down the road.