This paper continues an investigation into conformal structure on surfaces with boundaries that was initiated by \textit{X. Xu} and \textit{C. Zheng} [``Discrete conformal structures on surfaces with boundary (I) -- Classification'', Preprint, \url{arXiv:2401.05062}]. While it does offer a historical background, some definitions and a classification theorem from [loc. cit.] in its first two sections, it is quite difficult to follow this paper without first becoming well acquainted with the subject, as laid out in [loc. cit.]. As early as the first definition, the paper adopts a laxness concerning functional notation that is also an issue in [loc. cit.], and is perhaps an ``abuse of notation'' that is common and accepted in this field. Yet this is regrettable, as it makes it difficult for an outsider to follow what is being said. This first definition claims to be concerned with a function \(l : E \rightarrow \mathbb{R}\). But this cannot be correct if the rest of the definition is to make any sense. A careful and correct version of what is intended here can be found in [\textit{D. Glickenstein} and \textit{J. Thomas}, Adv. Math. 320, 250--278 (2017; Zbl 1379.52025)], along with a useful introduction to the whole subject. If we let \(\mathbb{R}^V\) and \(\mathbb{R}^{E^+}\) denote the vector spaces of real-valued function with domains \(V\) and \(E^+\), respectively, then the function \(l\) actually maps \(\mathbb{R}^V\) to \(\mathbb{R}^{E^+}\). Once this is realized, it becomes possible to make sense of the first definition in the paper. A topological surface with a boundary, here and in [Xu and Zheng, loc. cit], is presumed to have boundary components that are topologically circles. These components are each ``coned off'' to a point, and these points are used as the vertices in an ``ideal triangulation'' of the surfaces. In order to define a suitable concept of a ``discrete conformal structure,'' a ``partial lengths map'' \(d\) is introduced, which again should be a map from \(\mathbb{R}^V\) to \(\mathbb{R}^{E^+}\). This is required to satisfy \(l_{ij} = d_{ij}+d_{ji} > 0\), \(\sinh d_{ij} \sinh d_{jk} \sinh d_{ki} = \sinh d_{ji} \sinh d_{kj} \sinh d_{ik}\) and some partial differential equations. This ensures that the triangles in the triangulation, can be realized as triangles in the hyperbolic plane, and produce a reasonable Poincaré dual cellular structure. Sections 1.2 and 1.3 of [Glickenstein and Thomas, loc. cit.] motivates these requirements rather well. The goal of classifying such discrete conformal structures occupies the rest of [Xu and Zheng, loc. cit], and the results are stated as Theorem 1.4 there. The present paper reproduces this theorem (with a couple small changes) as their Theorem 3, and then proceeds to study some of the cases in the classification in greater detail.