H1B transfer and EAD and travel abroad

[Ira and Booboo]

I am currently employed on an H1B status (I have been on H1B since Feb 2010 with one employer B and switched to my current employer A in June 2012). My last day with my current employer A is on 9/6. I am currently in the process of visa sponsorship transfer back to my original employer B. I do not have a start date with B yet as the visa transfer process is pending. Employer B has made me an offer and I accepted and now they are in the process of filing the LCA (which they say will take about 10 business days to get completed). After that is done they will file the petition of H1B transfer. Tentative start date with B is 9/23 or 9/30.

In the same time I have an EAD card with the double purpose - as Employment authorization and as Advance Parole (I have this card through the green card application process that we initiated through my husband's employer).I already traveled outside of the US in May of this year and entered the US based on the EAD Advance Parole (on the I-94 homeland security stamped as 'Paroled'). I did not have an unexpired H1B visa at the time of entry. So I entered solely based on the EAD card.

Now we have family travel plans (already booked and paid for) outside of the US for the period of 9/13-9/22.

My question is - taking all of the above into consideration - can I travel outside of the US and safely return to the US based on the EAD card?

Thank you for your help.