We discuss structure formation in models with a spectator field in small-field inflation which accommodate a secondary period of inflation. In such models, subgalactic scale primordial fluctuations can be much suppressed in comparison to the usual power-law ΛCDM model while the large scale fluctuations remain consistent with current observations. We discuss how a secondary inflationary epoch may give rise to observable features in the small scale power spectrum and hence be tested by the structures in the Local Universe.