Magic wands doesn't work.
Tried using GIMP and zooming in to add to alpha channels but there must be a better way?
The only perfect result = repaint it on a transparent background. The shape isn't especially complex if you need only the same style.
One method to try in Photoshop:
Paint black to the layer mask on those areas that must absolutely be fully transparent. It's done here in top right corner. Paint white to the layer mask on those areas that should be fully opaque. Grey means partial transparency. White and grey were not added here.
Obviously much more painting into the mask is needed. That can be a little easier than repainting the image from scratch.
NOTE1: You can adjust layer mask if it's selected in the layers panel. You see its effect, not the mask. The mask itself pops onscreen if you click the mask icon holding Alt key at the same time. You need it when you paste the image to the mask.