A generalized inhomogeneous higher-order nonlinear Schrödinger (GIHNLS) equation for the Heisenberg ferromagnetic spin chain system in (1+1)-dimensions under zero boundary condition at infinity is taken into account. The spectral analysis is first performed to generate a related matrix Riemann–Hilbert problem on the real axis. Then, through solving the resulting matrix Riemann–Hilbert problem by taking the jump matrix to be the identity matrix, the general bright multi-soliton solutions to the GIHNLS equation are attained. Furthermore, the one-, two-, and three-soliton solutions are written out and analyzed by figures.