Overpainting can be done either by painting directly on the image texture in Image editor with a brush of the same color as the texture to manually create bleeding around UV islands or automatically with Bleed option.
Automatically this can be done in Texture paint mode, Options > Project Paint > Bleed (for 2.79). Increase it to 4 or 6 px. Then paint with those settings on the mesh, either with usual brush or with gradient paint (as your texture so far is one color).
Shrinking islands is done by scaling them. If there are multiple of them then try to use Individual origins as pivot point in UV editor, then scale (for very small amount). This might not work evenly if islands shapes are irregular. This method assumes texture is already drawn and won’t change; otherwise the method is practically useless as if baking or drawing with texture paint atop of shrunk islands the problem will appear again while UV islands will be smaller hence the texture resolution as well (a tiny bit but still).