<p>This paper concerns <i>BV</i> solutions of the one-dimensional Cauchy problem for a class of flux-saturated diffusion equations, where the flux saturates at large gradients, the initial data are discontinuous, and the source term may contain singularities. In contrast to most existing studies, which typically exhibit vertical solution profiles at points of discontinuity or restrict the propagation of the jump set to vertical lines, attention is given here to genuinely non-vertical jump discontinuity sets. By exploiting the structural properties of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(BV_x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <msub> <mi>V</mi> <mi>x</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> functions, the non-monotone flux can be consistently interpreted, and the problem is reformulated as a free-boundary problem governed by an entropy condition. Within this framework, explicit entropy solutions in <i>BV</i> are constructed that originate from discontinuous data, remain discontinuous throughout their evolution, and propagate a jump discontinuity along the curve, while preserving local classical regularity away from the interface. These solutions satisfy a Kruzhkov-type entropy condition but are not necessarily unique for the same initial data. Uniqueness of <i>BV</i> solutions is further established under the additional requirement that the derivative of the flux does not change sign on the jump set. The analysis yields a detailed characterization of the discontinuity dynamics and indicates how hyperbolic features may emerge in flux-saturated diffusion.</p>