You can't start working until you have your OPT work card in hand. There are circumstances where someone who hasn't yet graduated can be sponsored in the H-1B l

You can't start working until you have your OPT work card in hand. There are circumstances where someone who hasn't yet graduated can be sponsored in the H-1B lottery, but this essentially requires the student to have completed all requirements for the degree by the time of filing (in April).