Because of the one-dimensional confinement of electron momentum in narrow semiconductor wires, spin relaxation is suppressed irrespective of the presence of spin-orbit (SO) interaction. In quantum transport, weak localization corrections to conductivity are reflected as suppressed spin relaxation, which makes quantification of the SO strength difficult because quantum correction theory requires weak antilocalization when evaluating SO coefficients. Narrow wires with strong SO interaction are potential platform for Majorana particles and parafermions for topological electronics and quantum computation, so revealing the SO strength in semiconductor wire structures is beneficial. Herein, we present quantification of the full SO coefficient under weak localization in InGaAs-based narrow wires. Using anisotropic weak localization observed under an in-plane external magnetic field with various orientations, one can ascertain the relative ratio between Rashba (α) and linear Dresselhaus (β1) SO coefficients with no fitting. Furthermore, we find that widely tuning the potential profile of the quantum well through the top gate can expose a Rashba-predominant region in magnetoconductance, where the α value can be extracted reliably from two-dimensional quantum correction theory. Finally, we quantify the full SO coefficients including Rashba, linear Dresselhaus, and cubic Dresselhaus terms in the wire.