Guttman and Javier Thayer introduced the notion of unsolicited authentication tests, and used it to prove the correctness of security protocols in which a key server authenticate its clients. As an example, they have applied unsolicited authentication tests to prove the authentication goals of the Otway-Rees protocol. However, unsolicited authentication tests seem not to be fully explored in that case study, and the proofs were complicated. In this paper, we revisit the unsolicited authentication tests, and show how to strengthen and apply them in more general cases. To justify our work, we also use this extension to prove all agents' authentication guarantee of the Otway-Rees protocol.