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Don't you have Advance Parole? If so, you could leave and enter on THAT document.  You can work on your EAD or your H-1B. The following, hopefully, is helpful:

3. If an H-1 or L-1 nonimmigrant has traveled abroad and was paroled into the United States via advance parole, the alien is accordingly in parole status. Does this interim rule allow him or her to now apply for an extension of nonimmigrant status? 

Until the final rule is published, an alien who was an H-1 or L-1 nonimmigrant, but who was paroled pursuant to a grant of advance parole, may apply for an extension of H-1 or L-1 status, if there is a valid and approved petition. If the Service determines the alien's application for an extension of nonimmigrant status, the decision granting such an extension will have the effect of terminating the grant of parole and admitting the alien in the relevant nonimmigrant classification.

4. If an H-1 or L-1 nonimmigrant has traveled abroad and reentered the United States via advance parole, the alien is accordingly in parole status. How does the interim rule affect that alien's employment authorization?

A Service memorandum dated August 5, 1997, stated that an "adjustment applicant's otherwise valid and unexpired nonimmigrant employment authorization .is not terminated by his or her temporary departure from the United States, if prior to such departure the applicant obtained advance parole in accordance with 8 CFR 245.2(a)(4)(ii)." The Service intends to clarify this issue in the final rule. Until then, if the alien's H-1 or L-1 employment authorization would not have expired, had the alien not left and returned under advance parole, the Service will not consider a paroled adjustment applicant's failure to obtain a separate employment authorization document to mean that the paroled adjustment applicant engaged in unauthorized employment by working for the H-1 or L-1 employer between the date of his or her parole and the date to be specified in the final rule.

5. Should an alien returning to the United States from travel abroad who has a valid 1-512 and a valid H-1 or L-1 nonimmigrant visa be paroled in or readmitted in H-1 or L-1 status?

If an alien has a valid H-1 or L-1 nonimmigrant visa and is eligible for H-1 or L-1 nonimmigrant status and also has a valid Form I-512, he or she may be readmitted into H-1 or L-1 status or be paroled into the United States. It is the alien's prerogative to present either document at inspection. However, if an alien presents both a valid H-1 or L-1 nonimmigrant visa and a valid Form I-512, and the alien is eligible for the H-1 or L-1 nonimmigrant classification, the Service should inform the alien that H-1 and L-1 nonimmigrants no longer need to use advance parole to preserve pending applications for adjustment of status and should admit the alien in H-1 or L-1 nonimmigrant status. The fact that an alien has applied for advance parole and received Form I-512 does not compel him or her to use the advance parole.