Is there a different country that wants to get a bit jazzy with it?

Finding a country that will offer you a long-term visa to do "gig work" is really tough.

Visas work differently in every country, but one thing that is common to most countries is that a work visa is typically offered for an ongoing traditional job. Often it's limited to a salaried job offer.

Some countries will have special visa categories for musicians to come and do work in their country, but the few that I've heard of are offered on the basis of you coming to that country to work on a specific collaboration, or if you've been hired by a local organization for a specific event, or for a number of events over a period of time. Often a visa like this might be offered only for that period of time, and then you leave. But again, visas work differently everywhere, so this isn't universally true.

Aside from that, a more traditional work visa in most countries would be offered only if you have an actual job offer. So unless you find someone who wants to hire you as their full time "house jazz pianist", its not going to be easy to find a country that will give you a long-term visa (something that would let you really move there, rather than just a temporary stint).

If you're under the age of 35, probably the closest you will come to finding something like this is by looking into "working holiday" visas. These are visas that a few countries offer to citizens of specific other countries to come live temporarily in their country (sometimes for 6 months to 1 year), and they're often limited to people under 35. You can arrive without a job offer, and you can work freely in the country, or not work. I can't confidently say that all of them will tolerate "gig work", but some of them might.