One of the early influences on Newman's understanding of tradition was Edward Hawkins's book on unauthoritative tradition. Hawkins argues that God gave us not merely Scripture but the Church, and that both are legitimate means for attaining Christian truth. However, he is very clear that Scripture retains the dominant place. His argument for Church tradition is that it has presumptive value given that (1) the need for it is palpably felt when we try to teach Scripture; (2) it has in fact constantly existed; and (3) God has provided for it by instituting a system of ministers and teachers.
He thus sets up a sort of comparison between Scripture and the unauthoritative tradition of the Church. Scripture proves; Church tradition teaches. Scripture furnishes the foundation, Church tradition carries down the system. Scripture provides the substance, Church tradition provides the arrangement. Scripture demonstrates Christian truth, Church tradition by being unbroken points to it presumptively.
Hawkins, of course, is primarily thinking of catechesis, and his argument in favor of unauthoritative tradition is, in fact, an argument that the Church of England needs to do more to further its catechetical work. It's an argument that the catechesis of the Church has a fundamental role to play in spreading Christian truth. Interestingly, on such a view the Church becomes chiefly a catechetical system; Christians learn the truth from Scripture, and the purpose of the Church is chiefly to catechize people in it.
The emphasis on the word 'unauthoritative' serves the purpose of distinguishing this very Anglican view from the Catholic view. Catholics, of course, believe in unauthoritative tradition just as much as Hawkins. Hawkins, however, is very concerned to deny the 'Romish' view that there is a currently existing authoritative tradition. To have an authoritative tradition, Hawkins argues, you must have apostles, and we haven't had apostles in quite some time. Indeed, he suggests that there is a parallel between authoritative tradition and miracles: since he's a cessationist, he suggests that the transition from authoritative tradition to purely unauthoritative tradition is analogous to, and probably contemporaneous with, the transition from miracles to purely non-miraculous work. (Usually Newman's chief works on tradition are thought to be the book on the Arians and the essay on development; but, given that Newman was familiar with Hawkins's work, it's an interesting thought that perhaps the work on ecclesiastical miracles plays an essential role in Newman's thought and development on the subject of tradition. By reversing Hawkins and showing that the age of miracles did not cease, Newman can reverse the analogy, increasing the antecedent probability of an authoritative tradition.)
Hawkins also discussed the subject of unauthoritative tradition in his Bampton Lectures of 1840.