If the applicant does not have a filed H-1B petition within the 60 day grace period, can they continue to the stay within the US if their spouse is on a valid i

If the applicant does not have a filed H-1B petition within the 60 day grace period, can they continue to the stay within the US if their spouse is on a valid immigration status? Can the H-4 change of status be applied after this time while staying within the country?