We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP${}_n$s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations ($\propto ϵ$) in the Jordan-normal basis of the defective matrix at an EP${}_n$, we show that the upper-$k$ Hessenberg structure of the perturbation matrix directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto {ϵ}^{1/k}$, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP${}_3$s in P- and C-symmetric systems are restricted to at most $\sim {ϵ}^{1/2}$ branch points, while PT-symmetric systems generically support EP${}_3$s with the strongest possible singularities (viz. $\sim {ϵ}^{1/3}$). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band P- and C-symmetric models, establishing the existence of EP${}_4$s with $\sim {ϵ}^{1/4}$ and $\sim {ϵ}^{1/2}$ singularities, respectively.