A countable semigroup is ℵ-categorical if it can be characterised, up to isomorphism, by its first-order properties. In this paper we continue our investigation into the ℵ-categoricity of semigroups. Our main results are a complete classification of ℵ-categorical orthodox completely 0-simple semigroups, and descriptions of the ℵ-categorical members of certain classes of strong semilattices of semigroups.