We compute the exact and limiting smallest eigenvalue distributions for two classes of $β$-Jacobi ensembles not covered by previous studies. In the general $β$ case, these distributions are given by multivariate hypergeometric ${}_2F_{1}^{2/β}$ functions, whose behavior can be analyzed asymptotically for special values of $β$ which include $β\in 2\mathbb{N}_{+}$ as well as for $β= 1$. Interest in these objects stems from their connections (in the $β= 1,2$ cases) to principal submatrices of Haar-distributed (orthogonal, unitary) matrices appearing in randomized, communication-optimal, fast, and stable algorithms for eigenvalue computations \cite{DDH07}, \cite{BDD10}.