I don't think he can reenter the US as an unexpired F-1 visa stamp is alone not sufficient. He would have needed a valid unexpired I-20 with an endorsement from the school permitting the reentry to successfully use the F-1 visa to reenter the US.
I fear that the only two options would be to stay in Mexico (if the Mexican visa permits it) or to go back to home country and wait there.
Don't try to enter using F1 visa or any other questionable means. It'll make things worse. If your cousin works for a company that is generally legitimate, the H1 approval should come through. I think it's best you wait.
Do you know if the H1B approval came with a new I-94 attached at the bottom? If NO, then it may mean trouble. (Your cousin should have obtained a stamping before starting a job in that case).
I think the expense of consulting an attorney is not much more than a day's stay at a good hotel, so it may be worthwhile.
I fear that the only two options would be to stay in Mexico (if the Mexican visa permits it) or to go back to home country and wait there.
Don't try to enter using F1 visa or any other questionable means. It'll make things worse. If your cousin works for a company that is generally legitimate, the H1 approval should come through. I think it's best you wait.
Do you know if the H1B approval came with a new I-94 attached at the bottom? If NO, then it may mean trouble. (Your cousin should have obtained a stamping before starting a job in that case).
I think the expense of consulting an attorney is not much more than a day's stay at a good hotel, so it may be worthwhile.
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