A relaxed Steiner Tree Problem
Some observations:

First, let's take a different parameter called c. The problem is to find a minimum weight connected subgraph which connects at least c vertices in S. Now c=2 is the easy problem you refer to, and c=|S| is the Steiner tree problem.

I think for c=$|S|^\epsilon$, the problem is still NP-hard. The reduction would basically add lots of useless terminals connected to the rest of the graph with very high edge weights, so that these vertices will never be picked in the min weight tree, and the min weight tree will only have the vertices we wanted. For example, an algorithm to solve this problem for c=|S|/2 can be used to solve the Steiner tree problem by just taking the instance of Steiner tree, and doubling the number of terminals and connecting them to the rest of the graph with large edge weights.

As you mentioned, the c=2 case is in P. Similarly, for any constant c', the c=c' case is easy by a brute force search.