The ODE solver HBT(12)4 of order 12 (Can. Appl. Math. Q. 16(1) (2008) 77–94), which combines a Taylor series method of order 9 with a Runge–Kutta method of order 4, is expanded into optimal, one-step, 9-stage, explicit, strongstabilitypreserving (SSP), Hermite–Birkhoff–Taylor methods, HBT(p), of orders p = 6, 7, . . . , 12, with nonnegative coefficients. These methods are constructed by combining Taylor methods, T(p − 3), of orders p − 3 with a 9-stage Runge–Kutta method, RK(9, 4), of order 4. Several new one-step SSP methods arise with higher order than those appearing in the recent literature. The Shu– Osher form of RK methods is extended to the above combined methods. Compared to Huang’s k-step hybrid methods, HM(k, p), of the same order, the new HBT(p) generally have larger effective SSP coefficients and larger maximum effective CFL numbers on Burgers’ equation, independently of the number k of steps of HM(k, p). The new HBT(p) are listed in their canonical Shu–Osher form in the appendix.