Updating light gray area to darker a darker gray or black in an old photo

@patjenran Yes I'd use Levels too, I would do it in an 'adjustment layer, that way the Levels adjustment can be localised using a soft edge mask as @D Fosse suggested. With a layer you can also switch it on and off and alter it's opacity to reduce the effect you've created if need be . 

You’ve a pleasant task ahead. 

I would also be sure to archive a layered original TIFF or PSD file, only flattening resizing and sharpening for specific outputs at which point it's OK to make a Jpeg . 

I hope this helps
neil barstow, colourmanagement net - adobe forum volunteer - co-author: 'getting colour right'
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Yes, a Levels adjustment - but the key here is the mask. Use a large soft brush at low opacity to gradually build the adjustment where it's needed at the top and along the edges. Just hit x to paint with black or white in the mask.

I would try using Levels, to utilize the full range of brightness. I'd also add a Black and White layer, for the variations in color.