The concept of expansion of an information structure
You have to be careful what you can say is that if $\underline{S}$ be given by $(T^1,\pi^1)$ and $S$ is any information structure, say given by $(T^2,\pi^2)$, then there exists an expansion $S^*$ of $\underline S$ given by $(T^1\times T^2, \pi^1\cdot \pi^2)$ where $\pi^1\cdot \pi^2$ is the product distribution of $\pi^1$ and  $\pi^2$. Note that I have constructed the expansion, thus shown its existence.

The information structures, $S$ and $S^*$ are different since their support is in different spaces, $T^2$ vs $T^1\times T^2$. However, they are informationally equivalent. The equivalence here is in terms of the distribution of posterior beliefs induced by $S$ and $S^*$. Alternatively, if $\succsim$ is the Blackwell partial order, then $S\succsim S^*$ and $S^*\succsim S$. Your intuition, however, is correct, any information structure will be "essentially" an expansion of $\underline S$.