I had similar situation like yours. I was required to start my work before I got my OPT. That was two and half years ago. But I applied for three months CPT (Curriculum Practicum Training) from the International Students Office of our university; but I could not graduate at time because I was required to be a student to apply for CPT, and I registered for a 3-credit-hour online class and studied as a part time student. I am not sure if you have graduated or not. If not, you can check with your International Students office for this possibility. According to my knowledge, if you applied CPT for less than 6 months, you still can get one year OPT. And it is very easy to get you CPT, the director of you International students Office can approve for that, I didn’t need send paper work to Immigration Bureau at that time, I am not sure how they handle right now. Hope this is useful for you! Good luck!!