Chirality, defined by the absence of improper spatial symmetries, has attracted considerable attention as a symmetry principle that enables characteristic electronic, transport, and lattice responses, including current-induced magnetization (CIM) and chiral-phonon splitting (CPS). Recent symmetry-based approaches have shown that structural chirality can be characterized by electric toroidal (ET) multipoles, with the ET monopole $G_0$ and quadrupole $G_u$ playing central roles in cubic and noncubic systems, respectively. In this paper, we investigate achiral-to-chiral (AtC) structural phase transitions driven by atomic displacements and construct $G_{0,u}$ as explicit functions of the displacement order parameter $\eta$ using a group-theoretical approach. We show that the leading-order form of $G_{0,u}(\eta )$ is determined by the symmetry of the parent structure and the character of the displacive mode. This provides a symmetry-based classification of AtC transitions in which chirality is treated not merely as a binary property but as an induced structural order parameter with a characteristic dependence on $\eta$. Through a systematic analysis of various types of $G_{0,u}(\eta )$, we find $|G_{0,u}|\propto |\eta {|}^n$ with $n=1,2,3,\cdots ,$ including high-order multipolar chirality such as $n=$ 3, 4, and 6. We also discuss a distinct class of chirality, termed $k$-arm chirality, in which $G_{0,u}$ is formed from unequal invariant weights $|{\eta }_q{|}^2$ among different arms of the ordering wave vector $k$. We demonstrate that, when the response is governed by the ET-multipole coupling, the leading displacement dependences of CIM and CPS follow the same power law as that of $G_{0,u}(\eta )$. We further clarify the microscopic mechanism by which AtC transitions generate chiral phonons and the associated CPS, and show that folded optical modes can also exhibit nonchiral symmetry-lowering splittings whose displacement dependence differs from that of $G_{0,u}(\eta )$. Since $\eta$ evolves with temperature across the transition, the functional form of $G_{0,u}(\eta )$ governs the temperature dependence of ET-multipole-controlled CIM and CPS, providing a route to identify the symmetry of the underlying AtC transition.