Ruuth and Spiteri have shown, in 2002, that fifth-order strong-stability-preserving (SSP) explicit Runge-Kutta (RK) methods with nonnegative coefficients do not exist. One of the purposes of the present paper is to show that the Ruuth-Spiteri barrier can be broken by adding backsteps to RK methods. New optimal, 8-stage, explicit, SSP, Hermite-Birkhoff (HB) time discretizations of order p, p = 5, 6, . . . , 12, with nonnegative coefficients are constructed by combining linear k-step methods of order (p - 4) with an 8-stage explicit RK method of order 5 (RK(8, 5)). These new SSP HB methods preserve the monotonicity property of the solution and prevent error growth; therefore, they are suitable for solving hyperbolic partial differential equations (PDEs) by the method of lines. Moreover, these new HB methods have larger effective SSP coefficients and larger maximum effective CFL numbers than Huang's hybrid methods and RK methods of the same order when applied to the inviscid Burgers equation. Generally, HB methods combined with RK(8, 5) have maximum stepsize 24% larger than HB combined with RK(8, 4).