It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal polynomials also have surprising applications in the world of $q$. By inserting Chebyshev polynomials of the third and the fourth kinds into Bailey pairs, Andrews derived a family of Rogers--Ramanujan type identities and also results related to mock theta functions and Hecke--type series. In this paper, by constructing a new Bailey pair involving Chebyshev polynomials of the third kind, we further extend Andrews' way in the studying of Rogers--Ramanujan type identities. By fitting this Bailey pair into different weak forms of Bailey's lemma, we obtain a companion identity to Dyson's favorite one and also many other Rogers--Ramanujan type identities. Furthermore, as immediate consequences, we also obtain some results related to Appell--Lerch series and the generalized Hecke--type series.