My method for cutting an image out from a complicated background is to create a new layer and draw an outline on the inside edges of the object using the Line/Curve tool. Once that is complete I then fill the inside of that outline in using the Paint Bucket and then I use the Apply Texture Effect to redraw the original object over the new shape. The result is something like this:
This is a PNG image with a transparent background.
This JPEG image with a black background is to show how you can get
a clean cutout of the object without seeing those annoying white outlines
or blurred edges caused by using the Magic Wand Tool and Feather Effect.
I didn't bother to draw in the bottom piece but you get the idea.
Hope this helps.