We investigate whether a single-loop construction in non-adiabatic holonomic quantum computation can be generalized to multi-segment evolutions while preserving separation between geometric and dynamical contributions. Starting from a three-level Λ-system, we first study phase-only basis transformations between segments and find an exact non-trivial three-segment solution under correlated pulse-area and phase constraints. We then use a numerical loss-function approach to search for zero-leakage solutions with more than three segments, and find evidence of irreducible multi-segment evolutions in the tested low-segment cases. We further investigate more general unitary basis transformations and analyze the separation condition using projected dynamical generators in the computational subspace. The analysis suggests that admissible transformations must preserve a common dark direction, effectively reducing the problem to a two-dimensional bright-excited subspace. While an enlargement of the set of reachable single-qubit gates is not possible, the results may provide additional implementation freedom for realizing holonomic gates.