Vega-like stars are main-sequence stars that exhibit excess IR emission due to circumstellar dust grains which are probably distributed in discs. We have recently published an observational data base for a large sample of candidate Vega-like systems, comprising optical, near-IR and mm/submm-wave photometry, and mid-IR spectra. In a previous paper we presented radiative transfer models of eight sources from our sample that had low fractional excess luminosities. Here we present models of a further eight sources, all with large fractional excess luminosities dominated by excess emission at near-IR wavelengths. It was found that no single distribution of dust grains at thermal equilibrium in a disc could simultaneously match the excess emission at near-IR and longer wavelengths; We attempted to model the near-IR emission as due to thermally spiking small grains, which can temporarily attain the high temperatures required to produce excess near-IR emission. A near-ii spectrum of SAO 186777 shows the 3.3-mu m UIR emission band, confirming our earlier detection of UIR emission at longer wavelengths, and suggesting that small carbonaceous particles are responsible for some of the near-IR emission. The thermally spiking models were only partially successful and many of the sources required the presence of grains emitting in thermal equilibrium at similar to 1000-1500 K These grains must either be located very close to the stars (<1 au), or else be powered by accretion luminosity. Calculations of the optical depths of the model discs suggest the discs-are optically thick at visual wavelengths; optically thick modelling of these sources is desirable. The discs are optically thin at mm wavelengths, allowing us to confirm the presence of large grains in the discs. The stars presented in this paper may well be younger than the prototype Vega-like stars.