259. There are extreme circumstances in which not even the continuous rite can be celebrated. These occur when the danger of death from injury or illness is sudden and unexpected or when the priest is not called to exercise his ministry until the person is at the point of death.
The introduction to the Rite for Emergencies begins.
Some people still delay making the call to the parish for a priest or other minister. Many do wait until the moment of death–or even afterward. This is a relic of poor pre-conciliar catechesis–I wonder how the SSPX-ers handle it these days.
Trust my advice from the inside of the parish office and in many years of working with clergy: clue us in on a serious illness from the beginning. It’s not a bother or imposition; it’s our ministry.
260. In such a situation of emergency the priest should offer every possible ministry of the Church as reverently and expeditiously as he can. He may be able to provide only the barest minimum of sacramental rites and forms of prayer, but even then he should add other appropriate prayers to help the dying person and those who may be present.
Roman sensible shoes: do what can be done, when it can be done.