Massive stars with a core-halo structure are interesting objects for stellar physics and hydrodynamics. Using simulations for stellar evolution, radiation hydrodynamics, and radiative transfer, we study the explosion of stars with an extended and tenuous envelope (i.e. stars in which 95% of the mass is contained within 10% or less of the surface radius). We consider both H-rich supergiant and He-giant progenitors resulting from close-binary evolution and dying with a final mass of 2.8–5 M⊙. An extended envelope causes the supernova (SN) shock to brake and a reverse shock to form, sweeping core material into a dense shell. The shock-deposited energy, which suffers little degradation from expansion, is trapped in ejecta layers of moderate optical depth, thereby enhancing the SN luminosity at early times. With the delayed 56Ni heating, we find that the resulting optical and near-IR light curves all exhibit a double-peak morphology. We show how an extended progenitor can explain the blue and featureless optical spectra of some Type IIb and Ib SNe. The dense shell formed by the reverse shock leads to line profiles with a smaller and near-constant width. This ejecta property can explain the statistically narrower profiles of Type IIb compared to Type Ib SNe, as well as the peculiar Hα profile seen in SN 1993J. At early times, our He-giant star explosion model shows a high luminosity, a blue colour, and featureless spectra reminiscent of the Type Ib SN 2008D, suggesting a low-mass progenitor.