When should we call a priest for a dying relative?

Earlier than most families do. The most common reason a priest arrives too late is that nobody wanted to be the person who said it was time.

The Ethical and Religious Directives, the norms governing Catholic health care in the United States, say that all Catholics capable of receiving Communion should receive Viaticum when they are in danger of death, "while still in full possession of their faculties." Because Viaticum requires the person to be able to receive Communion, and because anointing normally comes before it, waiting for certainty tends to mean waiting too long.

Calling early costs nothing. Anointing can be repeated, and a priest who comes and finds the person stable will not be annoyed. A visit that turns out to be premature is a visit that reassured a frightened person.

Two practical steps. Ask, on a calm day, for the after-hours number for a Catholic priest at the hospital, and write it down in more than one place. If the person moves to a hospice or goes home, ask again, because the answer changes.

If you cannot reach anyone through the hospital, calling the nearest parish directly at night is entirely normal and expected.