This response is based on the information provided.  We suggest you discuss with your contracting and finance team, program manager and/or legal department as appropriate.

If the contract has expired, the most efficient way to get the original contractor to do additional work is a Justification & Approval (J&A) rather than conduct a full an open competition.  This assumes your statement is correct that the additional work is properly within the scope of the original effort.  If only the task order (TO) has expired, but the IDIQ contract is still in force, create another TO for the additional work.