In the slow-roll inflationary scenario, the amplitude of the curvature perturbations approaches a constant value soon after the modes leave the Hubble radius. However, relatively recently, it was shown that the amplitude of the curvature perturbations induced by the canonical scalar field can grow at super-Hubble scales if there is either a transition to fast roll inflation or if inflation is interrupted for some period of time. In this work, we extend the earlier analysis to the case of a non-canonical scalar field described by the Dirac-Born-Infeld action. With the help of a specific example, we show that the amplitude of the tachyonic perturbations can be enhanced or suppressed at super-Hubble scales if there is a transition from slow roll to fast roll inflation. We also illustrate as to how the growth of the entropy perturbations during the fast roll regime proves to be responsible for the change in the amplitude of the curvature perturbations at super-Hubble scales. Furthermore, following the earlier analysis for the canonical scalar field, we show that the power spectrum evaluated in the long wavelength approximation matches the exact power spectrum obtained numerically very well. Finally, we briefly comment on an application of this phenomenon. © IOP Publishing Ltd.