  We first study holomorphic isometries from the Poincar\'e disk into the
product of the unit disk and the complex unit $n$-ball for $n\ge 2$. On the
other hand, we observe that there exists a holomorphic isometry from the
product of the unit disk and the complex unit $n$-ball into any irreducible
bounded symmetric domain of rank $\ge 2$ which is not biholomorphic to any
type-$\mathrm{IV}$ domain. In particular, our study provides many new examples
of holomorphic isometries from the Poincar\'e disk into irreducible bounded
symmetric domains of rank at least $2$ except for type-$\mathrm{IV}$ domains.
