Time Complexity of a selection problem
If you are trying to prove $NP$-Hardness of that problem, you need a reduction from (not to).

Anyway, I don't think this is $NP$-hard (unless $P = NP$, of course)

Suppose the $S_i$ are sorted.

Suppose you decide that element $x \in S_j$ will be minimum. Now in each other set, you can find the smallest element greater (or equal to) $x$ using binary search, and take the maximum among those.

Do this for each of the $mn$ numbers, assuming it is the minimum. Pick the best set you get. This gives you a polynomial time algorithm: $m \log n$ time for each of the $mn$ elements, giving an $O(m^2n \log n)$ time algorithm.