New optimal strong-stability-preserving (SSP) Hermite–Birkhoff (HB) methods,  of order  with nonnegative coefficients, are constructed by combining -step methods of order  and -stage explicit Runge–Kutta methods of order 5 (RK5), where . These new methods preserve the monotonicity property of the solution, so they are suitable for solving ordinary differential equations (ODEs) coming from spatial discretization of hyperbolic partial differential equations (PDEs). The canonical Shu–Osher form of the vector formulation of SSP RK methods is extended to SSP HB methods. The  methods with largest effective SSP coefficient, , have been numerically found among the HB methods of order  on hand. These effective SSP coefficients are really good when compared to other well-known SSP methods such as Huang's hybrid methods (HM) and 2-step -stage Runge–Kutta methods (TSRK). Their main features are summarized.