Is my assumption correct? We would need to seek another method of making a contract award if we wish to continue with the requirement.
Once the contract options periods expire, the "contract can no longer continue" anyway, in that the basic contract will have expired. But to the central point of your question— Yes, you can continue to exercise the options. The following is from 13 CFR 125.18
(e) Recertification. (1) A concern that represents itself and qualifies as an SDVO SBC at the time of initial offer (or other formal response to a solicitation), which includes price, including a Multiple Award Contract, is considered an SDVO SBC throughout the life of that contract. This means that if an SDVO SBC is qualified at the time of initial offer for a Multiple Award Contract, then it will be considered an SDVO SBC for each order issued against the contract, unless a contracting officer requests a new SDVO SBC certification in connection with a specific order. Where a concern later fails to qualify as an SDVO SBC, the procuring agency may exercise options and still count the award as an award to an SDVO SBC.