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The primordial spectrum of fluctuations may present a large peak as a result of enhancing features during inflation. This may include, but is not limited to, bumps in the inflaton's potential, phases of ultra-slow-roll or turns in multi-field space. However, in many models, inflation does not end immediately after the enhancing feature and it is likely to continue with a second phase of slow-roll. We show that the resulting induced gravitational waves may probe the primordial spectrum from the second inflationary phase, even if its amplitude is too small to directly induce detectable gravitational waves. This is because, if there are sharp peaks in the primordial spectrum, the total gravitational wave spectrum is not simply the sum of gravitational waves induced by a peaked and scale-invariant primordial spectra separately, but cross terms from interaction between these modes also become important. We also find that such cross terms always have a characteristic slope. We discuss the parameter space that may be probed by future gravitational waves detectors in the presence of these signals.