@Cat-ophile, for context to comments then we are full-time international housesitters that travel overseas for extended periods :).
We never buy conventional travel insurance - as most coverage is not relevant for our needs. We never check luggage. Trip coverage only for first flight. Medical coverage is often short duration and very limited.
But we always purchase emergency medical insurance prior to departing home country. We have used two services (that offer multiple levels of coverage)-
- Genki that offers a flexible monthly subscription-based health insurance across many geographies
- Axa Schengen that overs Schengen+UK coverage for up to 180 days
To our knowledge, Genki does permit someone to start an insurance policy after they have left home country … but it applies limited coverage for first 14 days … presumably to mitigate risk of someone starting insurance coverage only after needing medical attention.