the main difference as you already mentioned is to switch from a local to a nationwide business. And it seems to me more a question of renaming instead of rebranding. If your business is running well your brand probably is well setup (positioning, value proposition, in other words the dna or essence of your brand).
Now it might be pretty obvious and you probably are aware of it, you need a name that appeals to a broader audience and you should definitely incorporate in your name the fact that you are a HEALTH insurance provider. Thus reworking all your website and digital material to optimize for keywords, tags to be found easily on the web. Going digital is a complete different pair of shoes from a walk in store, and I recomend to contract an expert in digital marketing. Hope that made sense. Feel free to come back to me if you´d like more advice. Kind regards, Joachim