If we consider $s$ as a variable, we can think of the affine family $H_{s}^{\operatorname{af{}f}}$ as one hypersurface given by a single equation in $(\mathbb{C}\backslash\{0\})^{d+1}$. Then the rescaled amoeba $\log|s|\cdot\mathcal{A}^{\lambda}_{s}$ sits inside the trace left by this extended $(d+1)$-dimensional amoeba in the horizontal hyperplane $\log|s|=const$. And the result follows from [GKZ94, Ch. 6, Prop. 1.9]. ∎