The integration of combustion chemistry into a fully compressible numerical solver is presently achieved using the hybrid transported-tabulated chemistry (HTTC). With HTTC, the main species are transported while most minor species are tabulated, which means that differences with a fully transported chemistry (FTC) solver are limited and concern mainly table reading for minor species. The implementation steps of HTTC are given in detail and an optimization of the code is proposed by tabulating the properties of the pure species as well as the reaction rates of the elementary reactions as a function of the temperature to speed up simulations. The original version of HTTC, validated for premixed combustion, has been also extended to partially premixed combustion by adding a prolongation of the lookup table for minor species outside the flammability limits. Two strategies are proposed and evaluated on a methane / air edge flame featuring a very high mixing fraction gradient. The results agree favorably by comparison with a reference flame simulated with a detailed chemistry. As the minor species are no longer transported with the flow using HTTC, the calculation cost is found divided by about 5 compared to the FTC solver.