We construct new one-step explicit multistage strong-stability-preserving (SSP) Hermite–Birkhoff–Taylor (HBT) time discretization methods of orders 3 to 5 for integrating hyperbolic conservation laws. The methods use derivatives y′ and y′′ as in Taylor methods of order two combined with Runge–Kutta (RK) methods of orders 2 to 4. Compared to RK methods of the same order and with the same number of stages, the new methods generally have larger SSP coefficients on Burgers’ equations. Moreover, these SSP HBT methods have stage order two, compared to stage order one for RK methods and hence are less susceptible to order reduction from source terms or nonhomogeneous boundary conditions.