At a guess, I would say that most of the name copywriters started out writing for a particular niche, got known, and then moved onto other areas.
I remember a teleseminar of Dr Kilstein's where he interviewed John Carlton. essentially, Carlton said that after a time, it doesn't really matter what business the client is in, when you've got it... you've got it.
(The difficult bit is the road to 'getting it')
As for Dr Kilstein he himself said that his earliest copy hits were for weight loss in his hipnosis clinic. That was undoubtedly his niche, so it was easy for him to transfer that into copy for the masses, as opposed to individual advice.
So, I would say that th easiest way to being a big name copywriter in all niches is to get known as a big name writer in any niche.
Get known as a health writer and a newsletter publisher will come knocking.
Get known as a real estate writer and a financial services company will come knocking.
But make sure you get known
When I say 'you' I of course mean the collective you.