Option one (going dense and bringing back the shadows) can lead to some issues I'd just as soon avoid. The biggie is the fact that when you try to bring up the darks (and blacks) you introduce noise into the image. It comes up fairly quickly, so often you'll have to decide between detail and noise.
Option two may, at first, seem to be the wrong way to go. After all, every new photographer in the digital age is cautioned not to over expose an image because once something is blown out the information is lost and there's no way to get it back. That is true, but it doesn't mean you can't push your histogram right up to the right hand edge. Today's image actually bumped up against the right edge. The sleeve closest to the camera was gone. There's a couple ways to deal with that, one good, one somewhat okay.
The upper side of the subject's hat appeared to be "off the charts". Nothing left to play with. Using the Adobe Photoshop Lightroom 5.5 (LR) (anything above version 4 will do) Adjustment Brush and a combination of the Whites and Highlights sliders resulted in what you see in the image. The detail was there. It just needed to be "developed" The braid was treated as a separate adjustment.
One "trick" involved a trip over to Adobe Photoshop (PS) because LR doesn't have the function needed. That function is Liquify. The whole time in PS was probably less than one minute. Once the image was brought over to PS the Liquify (Filter/Liquify) dialog box was opened. The Bloat Tool was use to enlarge the subjects right eye. No need to go crazy. Just a couple of quick taps of the mouse (or pen tool using a Wacom tablet) produces plenty of size. You don't want to make the change noticeable, just there. Back out of PS and Save/Close back over to LR. (Don't use Save As or the connection between LR and PS will be lost.)
The shoulder closest to the camera was in full sunlight and it was so blown out that all information was lost. Luckily I was shooting five shots at .3 F-stops jumps. I looked for the shot with detail on his shoulder. I Selected (CTRL clicked because they weren't serial ordered) the one I had been working on and the shot with the well exposed shoulder. I took them over to PS using the option "Open as Layers in Photoshop" (Photo/Edit/Open as Layers in Photoshop). Once there I reduced the Opacity of the top Layer to 50%. That way I could see through the Layer and check to see how well aligned they were. As it turned out, he had moved his head but not his shoulder. No problem. I put a Layer Mask on the top Layer and inverted the Mask (CTRL I [eye]). I used a half sharp white Brush (B) with the Flow set to 50% and painted over the shoulder. The blown out section was replaced by a properly exposed shoulder. No problem.
So, the moral of our story is to shoot with your exposure over to the right. I'd rather have the Histogram be away from the left (the dark side) rather than away from the right side (the bright side). You'll have an easier time (and not introduce noise) pulling the blacks down rather than needing to open them up.