The Beach Boys SMILE. Although the sessions didn't progress until 1967. Good Vibrations which was part of the SMILE sessions might have been released in 66' I forget. Smile some say was the answer to SGT. PEPPER. But that could be just a hypish concept. On the original tapes their voices are channeled through phase shifters and the vocals are very errie at times. The section which is known to be the Fire tapes is about 20 minutes long and has a bizzare reminiscence to the way Gentle Giant or King Crimson used tri tones. I know that sounds far fetched to most, but if I were to play the track you may not realize it's the Beach Boys your'e listening to.
The Lovin' Spoonful had strange songs. Booker T who was already mentioned and the Ventures.
There was a strange ballad back in the 60's titled: EVERYONE'S GONE TO THE MOON. The version I have is by Sara Hickman and it's a very strange sad little song. I am not sure who recorded the original.