High-speed solar wind streams (HSSs) interact with the preceding ambient solar wind to form stream interaction regions (SIRs), which are a primary source of recurrent geomagnetic storms. However, HSSs may also encounter and subsequently interact with interplanetary coronal mass ejections (ICMEs). In particular, the impact of the interaction between slower ICMEs and faster HSSs represents an unexplored area that requires further in-depth investigation. This specific interaction can give rise to unexpected geomagnetic storm signatures, diverging from the conventional expectations of individual SIR events sharing similar HSS properties. Our study presents a comprehensive analysis of solar wind data spanning from 1996 to 2020, capturing 23 instances where such encounters led to geomagnetic storms ( SymH < −30 nT). We determined that interaction events between preceding slower ICMEs and faster HSSs possess the potential to induce substantial storm activity, statistically nearly doubling the geoeffective impact in comparison to SIR storm events. The increase in the amplitude of the SymH index appears to result from heightened dynamic pressure, often coupled with the concurrent amplification of the CMEs rearward ∣ B ∣ and/or B _z components.