Your insurer may be able to give you a ‘green card’ that proves they will provide at least a minimum level of cover for you even once you leave the European Union. Ask your insurer if they can issue you with one.
If they can not then you will be responsible for obtaining local insurance at the border. You have a legal obligation to do so - after all, you wouldn’t expect a temporary visitor to your country to be driving around uninsured would you?
Depending on the border this will either be presented clearly at the point of immigration or you’ll have to hunt around the border for the relevant shack that has the regionally appointed insurance office. Fortunately in most countries outside the EU it is not terribly expensive - usually around $20 per country.