I question the very premise of your question.
Silent Image cannot do that. It permits "natural movement" of a creature while moving if you spend an action, not "infinitely thin holographic shell of a creature that mimics your movement".
Silent Image is not disguise self. You cannot generate a similarly flawless disguise this way. You also cannot create fake invisibility by projecting an illusion of the wall behind you that moves as you do; you could create an illusionary wall 3 feet away from the wall.
Having Silent Image move in exact sync with another object is not guaranteed to work by the spell. All you get is the ability to move it. You can try to keep it in exact sync, much like a dancer with two swords can try to move them so that the sword in front perfectly hides the sword behind, but it doesn't come for free, and doing so is a highly difficult task.
Such an act -- hiding someone in the shell of an illusion while letting them move freely -- would be an impressive stunt beyond the scope of what the spell guarantees you can do.
So the DM would have to invent rules to cover that stunt.
In a more general case, if you have an illusion of invisibility and under that you have disguise self, penetrating the invisibility illusion doesn't penetrate the disguise self.
If you have two magical effects that both generate the same effect, only the most powerful one takes effect. So if you have two different illusions that disguise you, the most powerful one works. This continues until the most powerful effect ends. So if you penetrate the most powerful effect in a way that doesn't end the effect (like seeing through it), the less powerful effect isn't active so is ignored. If you penetrate it in a way that ends it (like dispelling it), the less powerful effect takes effect.