We study the formation of superradiant bound states for massive scalar fields in five-dimensional rotating black string geometries with a non-vanishing Kaluza-Klein momentum along the compact direction. Even though all Kaluza-Klein modes may form bound states in this geometry, in realistic extra-dimensional models and astrophysical black holes only the zero-mode is sufficiently light for superradiant instabilities to develop, provided the field has a small but non-vanishing mass, as for example for axion-like particles. We use analytical and numerical methods to show that, although the Kaluza-Klein momentum decreases the upper bound on the field mass for an instability to develop, it may enhance its maximum growth rate by more than 50%, thus boosting the black hole bomb mechanism. We discuss the possible observational consequences of this result and its potential as an astrophysical probe of non-trivial extra-dimensional compactifications.