In this work, we present a generalized version of the semirigid vibrating rotor target (SVRT) model by including additional vibrational modes explicitly in the SVRT Hamiltonian. The inclusion of additional vibrational modes eliminates the uncertainty of fixing certain geometries of the target molecule as required in the basic SVRT model. This generalized SVRT (GSVRT) model was employed to study the benchmark reaction H+CH4 by including the umbrella mode of CH4. Influence of the umbrella mode of the reagent on reactivity is investigated. It is concluded that the inclusion of the umbrella vibrational mode of CH4 has only a small effect on the reaction from the ground state of the reagent, and essentially no effect from the excited C-H stretching vibrational state of the reagent. However, the initial excitation of the umbrella mode does give a sizable enhancement of reaction and reduces the reaction barrier by about 1.1 kcal/mol.