Generally speaking the IRS does not like to see an employee receiving both a 1099-Misc and a W-2 from one employer in the same year. But, in your situation that should not be a problem.
Now, to specifically address your question, if you have paid an independent contractor more than $600 in wages, then you are required to issue a Form 1099 Misc to the contractor, along with Red Copies mailed to the IRS of both Form 1096 and Form 1099-Misc (exactly the same copy issued to the Contractor). The 1099-Misc must be mailed to the contractor by January 31st usually, but since that date is on a weekend, the due date is Februray 2nd. However, the IRS due date for filing Form 1096 and Form 1099-Misc is generally due on February 28th but since it falls on a weekend, the due date is March 2nd!