A scanning challenge

I know several people on the forum are working on family pictures, some of which are very old. I've run up against a few that are really "challenging" me, and I'm hoping somebody has figured out a scanning trick that will not only cut down the amount of hand editing I have to do but also increase the quality.

A number of the old photographs I have that were taken in the early 1900s are still reasonably clear when you look at the originals, but after they've been scanned, the digital images have a lot of reflection in them. I think it's coming from the silver that was used in the processing, because the bad spots show up in the dark areas of the image - like in shadowed areas and on dark colored clothing. When looking at the originals "head on", you don't really notice it, but the reflective areas are very visible when you look at them on an angle, and they become downright grotesque once the light from the scanner gets done with them.

I've looked through Wayne Fulton's site on scanning. He covers issues of Moire and using various tools for correcting scratches, tears, etc., but I can't find anything on his site or anywhere else about how to minimize this annoying reflection. Cloning just smears everything to a blackish gray, and painting takes away any depth the original may have had. I've also fiddled with various filters, but I haven't found one that gets rid of the part I don't want without doing a lot of damage to what I want to salvage. There's got to be some technique I can use in the scanning stage that'll minimize this, doesn't there? Does anybody have any ideas? I'm open to experimenting, but I don't know what to try.