Can FAR 52.237-3, Continuity of Services, be modified into a contract and subsequently exercised on the same contract modification?
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.
This is a task order, so the first question is, "Is FAR Clause 52.237-3 in the IDIQ contract?" If so, it is easier to invoke. If not, the contractor does need to agree to the addition. Note that the clause if intended to extend the incumbent when the follow-on work has been awarded to another contractor and you need the incumbent to extend to affect a smooth transition.
If these conditions are not your circumstances, the appropriate approach is to pursue a Justification and Approval and clearly document the situation in a memo for record.