I think you'll find the answer you're looking for in the Caroll Spinney book, particularly in the section about Oscar's appearance on the Flip Wilson show.
Other than that I can offer this totally bogus, fake, untrue explanation that I made up, which will probably be interpreted to some in the Internet world as the truth.
Here goes:
Oscar's fur was always green from the get-go. During Woodstock in 1969, he had dyed it completely orange and stayed that way for all six months of Sesame Street's first season. When the show went into reruns, that was enough time for the green fur to grow out again and the orange fur to fall off or be cut off. So, when the second season began, there were his green fur.
(long pause)
Well, it could've happened that way.