This paper is the second in a series devoted to the study of spacetimes sourced by a stationary cylinder of fluid rigidly rotating around its symmetry axis and exhibiting an anisotropic pressure by using new exact interior solutions of general relativity. The configurations have been specialized to three different cases where the pressure is, in turn, directed alongside each principal stress. The two first articles in the series display the analysis of the axial pressure case. Indeed, the first axial class published in Paper I is merely a special case. It is recalled here and its properties are revised and supplemented. Moreover, a fully general method aiming at constructing different classes of such solutions is displayed. This method described in Paper II represents a key result of this work. It is exemplified and applied to two new classes of solutions depending on a single constant parameter. One of them, denoted Class A, is shown to verify every condition needing to be satisfied by a fully achieved set of exact solutions: axisymmetry and, when appropriate, regularity conditions; matching to an exterior vacuum; proper metric signature; and weak and strong energy conditions. Other properties and general rules are exhibited, some shedding light on rather longstanding issues. Astrophysical and physical applications are suggested.