Based on the results of detailed population synthesis models, Girardi et al. recently claimed that the clump of red giants in the colour-magnitude diagram (CMD) of composite stellar populations should present an extension to lower luminosities, which goes down to about 0.4mag below the main clump. This feature is made of stars just massive enough to have ignited helium in non-degenerate conditions, and therefore corresponds to a limited interval of stellar masses and ages. In the present models, which include moderate convective overshooting, it corresponds to ~1Gyr old populations. In this paper, we go into more detail about the origin and properties of this feature. We first compare the clump theoretical models with data for clusters of different ages and metallicities, basically confirming the predicted behaviour. We then refine the previous models in order to show the following behaviour. (i) The faint extension is expected to be clearly separated from the main clump in the CMD of metal-rich populations, defining a `secondary clump' by itself. (ii) It should be present in all galactic fields containing ~1Gyr old stars and with mean metallicities higher than about Z=0.004. (iii) It should be particularly strong, if compared with the main red clump, in galaxies that have increased their star formation rate in the last Gyr or so of their evolution. In fact, secondary clumps similar to the model predictions are observed in the CMD of nearby stars from Hipparcos data, and in those of some Large Magellanic Cloud fields observed to date. There are also several reasons why this secondary clump may be missing or hidden in other observed CMDs of galaxy fields. For instance, it becomes indistinguishable from the main clump if the photometric errors or differential absorption are larger than about 0.2mag. None the less, this structure may provide important constraints on the star formation history of Local Group galaxies. We comment also on the intrinsic luminosity variation and dispersion of clump stars, which may limit their use as either absolute or relative distance indicators, respectively.