I drove from Las Mercedes to Chamorga in June, it's really spectacular but the road seems to go on forever as it is very narrow and winding in places. The road ends at the Chamorga, which is a tiny village surrounded by mountains at the very tip of the island. There is actually another, even smaller village closer to the sea called Roque Bermejo but you can only get to it on foot, which is why I was there. It's a superb walk through the mountains and took around four hours.
There are a number of miradors along the road to Chamorga where you can pull over and admire the views. I returned via San Andres and picked up the motorway south at Santa Cruz.
I have driven the La Laguna/Teide route in the past and this is another spectacular drive. Allow yourself plenty of time and don't forget your camera.