Let G be a graph and S be a subset of V(G). A graph G is called S-leaf-connected if G has a spanning tree T such that S is the set of leaves of T. For k≥2, we say a graph G is k-leaf-connected if |V(G)|>k and given any subset S of V(G) with |S|=k, G is S-leaf-connected. In the present note, we give some sufficient conditions on eccentric connectivity index, eccentric distance sum, connective eccentricity index and difference of Zagreb indices for connected graphs to be k-leaf-connected. These results generalize the sufficient conditions on these indices for connected graphs to be Hamilton-connected. At last, we obtain a sufficient condition for a graph to be k-leaf-connected in terms of the general sum-connectivity index of the complement graph.