More than half of the sources identified by recent radio sky surveys have not been detected by wide-field optical surveys. We present a study, based on our co-added image stacking technique, in which our aim is to detect the optical emission from unresolved, isolated radio sources of the Very Large Array (VLA) Faint Images of the Radio Sky at Twenty-cm (FIRST) survey that have no identified optical counterparts in the Sloan Digital Sky Survey (SDSS) Stripe 82 co-added data set. From the FIRST catalogue, 2116 such radio point sources were selected, and cut-out images, centred on the FIRST coordinates, were generated from the Stripe 82 images. The already co-added cut-outs were stacked once again to obtain images of high signal-to-noise ratio, in the hope that optical emission from the radio sources would become detectable. Multiple stacks were generated, based on the radio luminosity of the point sources. The resulting stacked images show central peaks similar to point sources. The peaks have very red colours with steep optical spectral energy distributions. We have found that the optical spectral index αν falls in the range −2.9 ≤ αν ≤ −2.2 (), depending only weakly on the radio flux. The total integration times of the stacks are between 270 and 300 h, and the corresponding 5σ detection limit is estimated to be about mr ≃ 26.6 mag. We argue that the detected light is mainly from the central regions of dust-reddened Type 1 active galactic nuclei. Dust-reddened quasars might represent an early phase of quasar evolution, and thus they can also give us an insight into the formation of massive galaxies. The data used in the paper are available on-line at http://www.vo.elte.hu/doublestacking.