We investigate the emergence of rogue quantum amplitudes in discrete-time quantum walks (DTQWs) influenced by phase disorder. Our study reveals the statistics of occupation probability amplitudes in space and time, uncovering optimal disorder regimes that favor rogue wave events. Through numerical simulations, we demonstrate that the probability of rogue waves increases with quantum coins close to the Pauli-Z choice, regardless the disorder degree. Conversely, for coins near Pauli-X rogue events are scarce, except under weak disorder. A monotonic threshold is observed between rare- and high-probability rogue wave regimes, depending on the quantum coin. We provide a comprehensive analysis of the coin-disorder interplay to rogue wave events. Our findings shed light on the possible control of extreme quantum amplitudes through quantum coins in disordered DTQWs.