Stability switchings of linear time-invariant (LTI) multiple time-delay systems arise on some hypersurfaces defined in the delay parameter space. These hypersurfaces, known as potential stability switching hypersurfaces (PSSH), can be generated from core hypersurfaces (CH) found in pseudo-delay domain using the Rekasius transformation within the cluster treatment of characteristic roots approach. In this paper, we present some interesting features of CH, with which the detection of PSSH can be simplified, and analyse, departing from the formulation of CH, the asymptotic directions of PSSH at infinity, which can be important in robustness and fragility analysis. We achieve these by constructing an algebraic approach that requires studying single-variable polynomials. Case studies are provided to demonstrate the developed approaches.