Turning white pixels transparent and getting clean edges in Photoshop CS3

I have a gif with a white background, and I want to turn all the white pixels in that gif transparent.  I followed the advice of the accepted solution in this thread:

https://www.experts-exchange.com/questions/23220932/How-do-change-every-white-pixel-in-an-image-into-a-transparent-pixel-with-photoshop-CS3.html

Unfortunately there were off-white pixels that stood out against non-white backgrounds, making the whole thing look messy.  Does anyone have any suggestions as to how I can clean up these edges?

Thanks