Sorry this is days later, but no you don't. Like I said, Medium Deeper Wells unlocks Hard, which unlocks insane. Then after you do Deeper Wells on insane you can go right to Ancient Mines on insane without ever having done Medium or Hard Ancient Mines. I know, this is what I did.
You can skip medium and hard difficulties. On those difficulties beating a map unlocks that map on the next difficulty. So you can go Medium Deeper Wells > Hard Deeper Wells > Insane Deeper Wells. I believe you have to clear insane (and be level 70) to unlock Nightmare though.