You’ve hit upon one of the dilemmas photographers encounter on a regular basis and this dilemma, most likely does not have a good solution. White or gray overcast skies (or very hazy skies) lack any kind of appreciable detail that you use compositionally. Nor does the featureless sky have enough detail that you can enhance, HDR or otherwise. Another issue caused by overcast skies is the flattening of light. Shadows will disappear and details in various objects smooth. Lighting that is perfect for shooting portraits outdoors, but not so good for a subject that may require strong dramatic lighting to make the image work. So what is the solution? I would argue that there is no technical fix to this problem, but one should instead see and shoot differently in order to use the available light to your benefit and not fight against it. Look at doing close up detail shots, or portraits where you can minimize the impact of the featureless sky by controlling the depth of field or eliminating altogether by framing your image.