How do retiree health benefits work when you move out-of-state?
As you stated, Blue Cross/Blue Shield is a supplemental health care plan to Medicare. It is sometimes called Medigap insurance because it fills in the gaps left by Medicare Parts A and B, e.g. deductibles, copays, etc. However, neither Medicare nor Medigap pay for long term care in an ALF -- Assisted Living Facility. They strictly pay for medical expenses involving skilled care, i.e. nursing or therapy under a doctor's orders.
Even though your mother's health care plan covers her medical needs anywhere in the country, she will have to find other means to pay for assisted living. The only insurance that covers assisted living is long term care insurance. Most newer policies cover assisted living and home care. However, many older policies are restricted to nursing home coverage only. If your mother has a long term care policy, check with her agent or the issuer to determine what her benefits cover.
