The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets

Let $M$ be the matrix whose rows are the vectors $\boldsymbol{h_i}$. Then the $r$-dimensional volume of $\mathcal{S}=\mathcal{S}_1+\cdots+\mathcal{S}_n$ is equal to the sum of the absolute values of the $r\times r$ minors of $M$. I don't know who originally showed this, but one can show that $\mathcal{S}$ can be tiled with $r$-dimensional parallelotopes whose volumes are the $r\times r$ minors. This follows for instance from the proof of Lemma 2.1 here.

The keyword you are looking for is "zonotope", which is defined to be the Minkowski sum of line segments. An early reference for zonotope is: P. McMullen, “On zonotopes”, Transactions of the American Mathematical Society, Vol. 159, 1971.

Following your notation, the $r$-dimensional volume of the zonotope $\mathcal{S}_{1} + ... + \mathcal{S}_{n}$ is equal to

$$\displaystyle\sum_{1\leq i_{1} < i_{2} ... < i_{r}\leq n} \big\vert{\rm{det}}\left(\mathbf{h}_{i_{1}},\mathbf{h}_{i_{2}},...,\mathbf{h}_{i_{r}}\right)\big\vert.$$

For reference, see eqn. (57) in "Combinatorial Properties of Associated Zonotopes" by G.C. Shephard, Canadian Journal of Mathematics, 1974. In that paper, there is an extra factor $2^{r}$ in front of the above expression since the line segments there are defined by $\{a\mathbf{h}_{i} : -1\leq a \leq 1\}$ instead of the OP's convention: $0\leq a \leq 1$. In the very end of this paper, Shephard credits McMullen for drawing attention to this formula. The same formula also appears as Exercise 7.19 in G.M. Ziegler, Lectures on Polytopes, Vol. 152, Springer, 2012; screenshot below: