What is the quickest Mac OS X app to use to remove the background of an image to make it transparent?

I followed the instructions in How to use Preview for Basic Image Editing: Remove Backgrounds and Extract Shapes to use the Preview app to remove backgrounds with an image. The problem is that the "Instant Alpha" tool sometimes does not capture all the pixels and can't figure out how to use the rectangle tool to remove those remaining pixels. I tried highlighting the pixels and then pressing delete but it doesn't work. Can someone explain how to do it or recommend another free, lightweight Mac OS X tool to remove background to make it transparent? I do not want to use Photoshop or anything else expensive.