Reducing infinite paths of a transition system to its set of sets of states
Since the set you want to output can be exponential in the size of the transition system, there is clearly no polynomial algorithm to enumerate it.
Thus, let's focus on an exponential time solution. To this end, it is actually quite easy to find a naive algorithm:
For every subset $Q\subseteq S$ (that contains $s_0$), in order to check whether $Q$ should be outputted by the algorithm, look at the subgraph induced by $Q$, and check whether there is a path from $s_0$ that goes through all the vertices in the induced subgraph.
One way to do the latter, is to indeed decompose the subgraph to its strongly connected components (SCCs), then look at the tree of SCCs, and make sure it's simply a directed path from $s_0$. If it is not a path, then you cannot traverse the entire set $Q$, as you would need to select a branching in the tree from which you cannot go back.
Thus, the test of whether $Q$ should be outputted can be done in polynomial time.