Based on this advice from UC Berkeley, there is no late start grace period.
Be aware of your grace periods.
You have a 60-day grace period at the end of your F-1 program. The grace period ends when you depart the United States, or no later than 60 days after the end of the I-20 or your EAD card (if on OPT), whichever comes first.
An F-1 student may enter the United States initially up to 30 days before, but no later than, the program start date on the I-20. (Emphasis added)
An F-1 student may remain in the United States for up to 60 days beyond:
- the completion of the program of study
- the completion date of any authorized post-completion OPT, including STEM OPT and Cap Gap.