A right \(R\)-module \(M\) over a right Noetherian ring \(R\) is \(\beta\)-stable (for an ordinal \(\beta\)) if the set of finitely generated submodules of \(M\) with Krull dimension less than \(\beta\) is closed under essential extensions. In [A. V. Jategaonkar, Trans. Am. Math. Soc. 190, 109- 123 (1974; Zbl 0284.16003)] it was proved that all modules over a right and left FBN ring are \(\beta\)-stable (for any \(\beta\)). The theory of rings with the second layer condition (s.l.c.) (which has its roots in the above paper) enables the above result to be easily extended to tame modules over a Noetherian s.l.c. ring for which certain symmetry conditions on Krull dimension are satisfied. In the present paper these ideas are developed in a setting which permits the extension to right Noetherian s.l.c. rings of [J. A. Beachy, J. Pure Appl. Algebra 24, 235-244 (1982; Zbl 0489.16019)] (in which Jategaonkar’s result was generalized to a characterization of \(\beta\)-stable modules over right FBN rings).