Remember that light bounces
Light bounces on all surfaces. If there’s only one light source in the scene and the ground underneath the object is lit by sunlight, it then becomes a secondary light source that I need to take into account. Less bright than direct light, reflected light from the ground will appear in the core of the shadow and lighten its tonal values. Bear in mind that the intensity of the secondary light source is lower than that of the main one.
Improving the forms
I keep rendering the shadow side because I feel that it’s a bit too light, adding a Multiply layer on top of the ship and refining the shadow shapes. I also use the Blend If function (see Pro Secret below) to control this Multiply layer. I’m trying to be subtle so I lower the Opacity of the layer blending modes, especially with dark areas. It gives me more control over values.