What outpatient therapies do you recommend for continuous or severe pain from migraine that is refractory to acute therapies?

I assume that what we are dealing with here is what we call a migraine status or status migrainosus. This refers to a migraine headache that has continued well beyond its usual duration, for which we often use the arbitrary three-day mark. We base this mark on the erroneous assumption that migraine headaches last anywhere from 4 to 72 hours, which is total nonsense. In general, migraine headaches minimally last a good part of the day, if not the whole day, requiring overnight sleep to be fully relieved, especially in adults. There is no maximum duration and I personally consider it a migraine status if the headache, as aforementioned, continues well beyond its usual duration for the particular patient, whatever that is.

An effective treatme...

I agree with Dr. @Dr. First Last that the two best medications to stop status migrainosus or any long-duration migraine are dihydroergotamine (DHE) and corticosteroids.

It is ideal when the patient can give themselves a subcutaneous injection of DHE at home, but this is difficult to arrange and for the patient to do. It will be wonderful when a new Drug/Device combination is released later this year. Hopefully, that enables the patient to do a self-injection of subcutaneous DHE at home.

For corticosteroids, I prefer dexamethasone 4 mg tablets, and I usually use three of them the first day, two the next day, and one the following day. Sometimes a sleeping aid like zolpidem can be helpful.

Once status migrainosus (a migraine attack lasting more than 72 hours) sets in, oral acute medications usually do not work. While some people have effectively used a neuromodulation device (Nerivio), studies included patients with headaches lasting <72 hours, so its efficacy for status migrainosus remains uncertain.

Intranasal DHE can be another potential option, particularly for individuals who have responded positively to IV DHE in the past.