Static or frozen disorder, characterised by spatial heterogeneities, influences diverse complex systems, encompassing many-body systems, equilibrium and nonequilibrium states of matter, intricate network topologies, biological systems, and wave-matter interactions. While static disorder has been thoroughly examined, delving into evolving disorder brings increased intricacy to the issue. An example of this complexity is the observation of stochastic acceleration of electromagnetic waves in evolving media, where noisy fluctuations in the propagation medium transfer effective momentum to the wave. Here, we investigate elastic wave propagation in a one-dimensional heterogeneous medium with diagonal disorder. We examine two types of complex elastic materials: one with static disorder, where mass density randomly varies in space, and the other with evolving disorder, featuring random variations in both space and time. Our results indicate that evolving disorder enhances the propagation speed of Gaussian pulses compared to static disorder. Additionally, we demonstrate that the enhanced speed effect also occurs when the medium evolves chaotically rather than randomly over time. The latter establishes that evolving randomness is not a unique prerequisite for observing the enhanced transport of wavefronts, introducing the concept of chaotic speed enhancement of waves in complex media.