Maximizing Opioid Recovery Emergency Savings Act or the MORE Savings Act This bill modifies coverage of opioid treatments and recovery support services under Medicare, Medicaid, and private health insurance. Specifically, the bill requires the Center for Medicare and Medicaid Innovation to test a model in which specified opioid treatments and recovery support services are provided under Medicare without cost-sharing (e.g., coinsurance, copayments, and deductibles). The bill also allows state Medicaid programs to cover recovery support services as part of medication-assisted treatment (MAT) and increases the applicable Federal Medical Assistance Percentage for MAT. Private health insurers must also cover specified opioid treatments and MAT-associated recovery support services without cost-sharing.