This isn't really a proper answer to your question, but is a bit too long for a comment.
The quantity you are after will vary from graph to graph, and will depend on the initial site of the walker. The expected number of distinct intermediate nodes will depend strongly on clustering within the graph, and I would expect the expected number of distinct intermediate nodes to be correlated with the clustering coefficient.
A cluster is basically a subset of vertices which share a large number of edges, so that each vertex is connected to a large fraction of the other vertices within the cluster. When a walker enters a cluster it is likely to stay in that region for a large number of hops, possibly revisiting each node many times. Indeed, using random walks in this way is one of the computational techniques used for identifying clusters in large graphs. Thus for a walker starting in a cluster, the expected number of distinct intermediate vertices will likely scale with the size of the cluster and the average probability of leaving the cluster.
While the above applies to both weighted and unweighted graphs, lets take the maximally connected unweighted graph of $N$ vertices as an example. In this case, at each hop after the first, the walker has probability $\frac{1}{N}$ of returning to the initial vertex. Thus the expected number of hops to return to the initial site is $N+1$. Even if we connect some vertices in this graph to a vertex in some other graph (i.e. outside of this clique) the probability of each hop leaving the cluster before returning to the initial site can be very low. Thus we expect clustering to reduce the number of distinct intermediate vertices by confining the walker within the cluster.
The average degree of vertices within the graph will also play an important role, though this is linked to the clustering. The reason for this is that when the walker jumps onto a vertex with degree 1, it must hop back to the previous vertex on the next hop. Even when the degree is 2, there is only one path which can be followed through the graph, although it can be traversed in either direction at each hop. On the other hand, for graphs with degree higher than 2, the number of paths can explode, making it extremely unlikely to return to the initial site even if the shortest path between then is small.
Thus you would expect the number of distinct intermediate vertices to be high for graphs which both have an average degree substantially above 2, and also have no significant clustering, such as trees.
Of course these comments no longer hold in the case of quantum random walks, but I guess you only care about the classical case.