Honestly, I think either idea would work. But, personally, I would build off of the already established website.
Reason being this; if you were to make a new site - and it was optimized correctly, had amazing user-friendly content, and was all around a great website - It wouldn't matter what the DA is, it would get ranked. I've seen it happen plenty of times where a new site comes onto the scene and ends up taking over well-established sites in the rankings, and it boils down to having a better, more user-friendly website.
But, like I said earlier, I would attach these new pages to your old domain. If you build these pages to be just as amazing and user-friendly and useful as you would a new website, those pages will still rank. And not only will that help getting ranked for those specific search terms, it will build the credibility of your root domain and could boost your DA.
Hope this helps!