If the task is non-severable, can the work go beyond the contract PoP? If yes, is there a "drop dead date"? (The proposal has the effort going 20 months beyond!) If no, what are the options for the effort?
Once a contract has expired because the period of performance has passed, there is no longer a contractual relationship. Generally speaking, the authorities within a contract where this can be an issue, such as the Changes clause and Option to Extend the Term of the Contract clause, are no longer in effect. Extending the relationship with the contractor at this point would essentially be a sole source award of a new contract, and would require a Justification and Approval in accordance with FAR 6.3. FAR 17.204(d) permits the period in which an option may be exercised to extend beyond the contract completion date for services, but that does not apply in your scenario because no options are involved. You will need a bilateral modification to extend the period of performance (in addition to the J&A to extend the POP).
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I recommend that you check with your organization's legal counsel for an authoritative response, as the legal issue of what constitutes an ongoing contractual relationship is an important one. Any "drop dead date" may then depend on funding issues (e.g., bona fide need considerations, expiration of funds).