The methods of Invisible Invariants and Invisible Ranking were developed originally in order to verify temporal properties of parameterized systems in a fully automatic manner. These methods are based on an instantiate-project-and- generalize heuristic for the automatic generation of auxiliary constructs and a small model property implying that it is sufficient to check validity of a deductive rule premises using these constructs on small instantiations of the system. The previous version of the method of Invisible Ranking was restricted to cases where the helpful assertions and ranking functions for a process depended only on the local state of this process and not on any neighboring process, which seriously restricted the applicability of the method, and often required the introduction of auxiliary variables. In this paper we extend the method of Invisible Ranking to cases where the helpful assertions and ranking functions of a process may also refer to other processes. We first develop an enhanced version of the small model property, making it applicable to assertions that refer both to processes and their immediate neighbors. This enables us to apply the Invisible Ranking method to parameterized systems with ring topologies. For cases where the auxiliary assertions refer to all processes, we develop a novel proof rule which simplifies the selection of the next helpful transition, and enables the validation of the premises possible under the (old) small model theorem.