H1b->quit->school (F1). Got new job in F1 grace period. Ok to file H1b transfer COS?

[aka0908]

I got H1b from my first job in U.S. Then I quit and went to school with F1. Recently I found a new job but I'm already in F1 60-day grace period. Can I file H1b transfer COS instead of consular processing? If H1b transfer COS is filed before the grace period ends but the grace period ends before H1b transfer is approved. Do I still have legal status in U.S?

I know people can file h1b transfer COS while under H1b grace period. But I'm in F1 grace period. I'm not sure if I can file COS without leaving the country or I have to do consular processing.

Thanks for any input!