Context. The quasar-type blazar 3C 454.3 was observed to undergo an unprecedented optical outburst in spring 2005, affecting the source brightness from the near-IR to the X-ray frequencies. This was first followed by a millimetric and then by a radio outburst, which peaked in February 2006. Aims. In this paper we report on follow-up observations to study the multiwavelength emission in the post-outburst phase. Methods. Radio, near-infrared, and optical monitoring was performed by the Whole Earth Blazar Telescope (WEBT) collaboration in the 2006-2007 observing season. XMM-Newton observations on July 2-3 and December 18-19, 2006 added information on the X-ray and UV states of the source. Results. The source was in a faint state. The radio flux at the higher frequencies showed a fast decreasing trend, which represents the tail of the big radio outburst. It was followed by a quiescent state, common at all radio frequencies. In contrast, moderate activity characterized the near-IR and optical light curves, with a progressive increase of the variability amplitude with increasing wavelength. We ascribe this redder-when-brighter behaviour to the presence of a ""little blue bump"" due to line emission from the broad line region, which is clearly visible in the source spectral energy distribution (SED) during faint states. Moreover, the data from the XMM- Newton Optical Monitor reveal a rise of the SED in the ultraviolet, suggesting the existence of a "" big blue bump"" due to thermal emission from the accretion disc. The X-ray spectra are well fitted with a power- law model with photoelectric absorption, possibly larger than the Galactic one. However, the comparison with previous X-ray observations would imply that the amount of absorbing matter is variable. Alternatively, the intrinsic X-ray spectrum presents a curvature, which may depend on the X-ray brightness. In this case, two scenarios are possible. i) There is no extra absorption, and the X-ray spectrum hardens at low energies, the hardening being more evident in bright states; ii) there is a constant amount of extra absorption, likely in the quasar environment, and the X-ray spectrum softens at low energies, at least in faint X-ray states. This softening might be the result of a flux contribution by the high-frequency tail of the big blue bump.