There are a several Mids out there that give (virtually) the same ankle support as High-tops, the real advantage of a Mid or a low is less weight and more articulation ( its a lot easier to drive a vehicle in a mid/low too ! )
I guess that shoe weight, trail type and your own personal ankle roll/injury likeliness should weigh the final shoe choice.
For Me, I usually hike in tall-Mids or Highs to give the support I need to carry a fairly large pack and off-trail I have a bit more protection from low-lying cactus and ankle-biting vermon (aka snakes)....Yeah, plus I`ve already rolled an ankle BAD with a loosley-tied Mid so I am a bit more paranoid about re-injury now.
Peace and happy hiking !!