This preprint develops the first canonical local refinement of the odd branch in hierarchy-compressed transport. For odd local order m, the turning-point configuration is of crossing type. Once the distinguished decaying branch is fixed on the evanescent side, continuation across the turning point determines a canonical local crossing datum \mathcal C_m=(\Phi_m,A_m), consisting of a canonical local crossing phase and amplitude coefficient. The first two odd hierarchy levels, m=1,\qquad m=3, thus carry the first canonical odd-branch continuation data \mathcal C_1 and \mathcal C_3. This note remains strictly local and defers extraction machinery, semi-global laws, even-branch canonical structures, and physical matching problems.