For PNGs, one way is to use a graphics program (Photoshop, Gimp, Paint.NET, etc) to add a mask around the edge, either a crisp one that cuts off the offending "outline" or a gradient one that fades outward from the image and softens the outline so it isn't so harsh.
Another approach (or one you might use in addition) might be antialiasing, or using a blur or soften effect around the edge.
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Another approach (or one you might use in addition) might be antialiasing, or using a blur or soften effect around the edge.