  We show that in algebraically locally finite countable homogeneous structures
with a free stationary independence relation the small index property implies
the strong small index property. We use this and the main result of [15] to
deduce that countable free homogeneous structures in a locally finite
relational language have the strong small index property. We also exhibit new
continuum sized classes of $\aleph_0$-categorical structures with the strong
small index property whose automorphism groups are pairwise non-isomorphic.
