A Riemannian manifold $\left( M,\,\rho \right)$ is called Einstein if the metric $\rho $ satisfies the condition $\text{Ric}\left( \rho \right)\,=\,c\,\cdot \,\rho $ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $SO\left( n \right)/SO\left( l \right)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $SO\left( n \right)/SO\left( l \right)$ that admits at least $p$$SO\left( n \right)$-invariant Einstein metrics.