For a vacation in the southwest USA, we're considering renting a high-clearance vehicle to visit locations such as the Toroweap overlook (also known as Tuweep overlook). Regular car rental classifications appear to classify cars as "Small", "Medium", "Large", "SUV" (all cheap), "Luxury", "Van", or "Convertible" (all expensive). I don't know if any of those may be expected to be high clearance. Independent companies rent actual Jeep Wranglers but judging from cars I've seen on similar high clearance roads in the past, a car like the Jeep Wrangler is overkill. How can I specifically rent a high clearance vehicle, without going all the way to specialised jeep rentals which would triple the rental price?
NB: See also this sister question on whether terms and conditions would allow driving on unpaved dirt roads at all.