For any a matrix $X$, based on the Cauchy Interlacing Inequality below, I guess that \begin{equation} \sigma_{\min}(X) \leq \sigma_{\min}(Y), \end{equation} where $Y$ is any a sub-matrix of $X$. I don’t know if this guess above is correct.
Cauchy Interlacing Inequality: Given any Hermitian matrix $A \in \mathbb{C}^{n \times n}$ and column orthonormal $U_k \in \mathbb{C}^{n \times (n-k)}$ with $1\leq k \leq n-1$, we have \begin{equation} \lambda_{\min}(A) \leq \lambda_{\min}(U_k^HAU_k) . \end{equation}
Let $X=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix},$ then $X$ is an orthogonal matrix and $\sigma(X)\equiv 1.$ But there's many submatrix $Y$ with $\sigma_{\min}(Y)=0,$ even if you choose $Y$ diagonally.