Let $T(S)$ be the Teichmüller space of a hyperbolic Riemann surface $S$. Suppose that $\mu $ is an extremal Beltrami differential at a given point $\tau $ of $T(S)$ and $\{ {\phi }_{n} \} $ is a Hamilton sequence for $\mu $. It is an open problem whether the sequence $\{ {\phi }_{n} \} $ is always a Hamilton sequence for all extremal differentials in $\tau $. S. Wu [‘Hamilton sequences for extremal quasiconformal mappings of the unit disk’, Sci. China Ser. A 42 (1999), 1033–1042] gave a positive answer to this problem in the case where $S$ is the unit disc. In this paper, we show that it is also true when $S$ is a doubly-connected domain.