I think you could start applying now since you already know your social security number.
Most firms require you to fill the I-9 form on the orientation day if you're offered a position. A copy of your valid EAD is sufficient for the I-9. You don't need to submit a copy of your social security card.
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|Date of initial EAD| 4/1/2013, expire 3/31/2015
|1st renewal| expire 3/6/2017
|2nd renewal| expire 3/6/2019