The Survey for Distant Solar Twins aims to find stars very similar to the Sun at distances of 1-4kpc, several times more distant than any currently known solar twins and analogues. The goal is to identify the best stars with which to test whether the fine-structure constant, {alpha}, varies with dark matter density in our Galaxy. Here, we use EPIC, our line-by-line differential technique, to measure the stellar parameters - effective temperature Teff, surface gravity log g, and metallicity [Fe/H] - from moderate-resolution (R <~ 32000) spectra of 877 solar twin and analogue candidates (547 at 1-4kpc) observed with the High Efficiency and Resolution Multi-Element Spectrograph (HERMES) on the Anglo-Australian Telescope. These are consistent with expectations for Teff and log g from photometry, and for [Fe/H] from the Besancon stellar population model. EPIC provides small enough uncertainties (~90K, 0.08dex, and 0.05dex, respectively), even at the low signal-to-noise ratios available (S/N ~> 25 per pixel), to identify 299 new solar analogues (=> 90 per cent confidence) and 20 solar twins (=> 50 per cent confidence), 206 and 12 of which are at 1-4kpc. By extending EPIC to measure line broadening and lithium abundance from HERMES spectra, and with ages derived from isochrone fitting with our stellar parameters, we identify 174 solar analogues at 1-4kpc that are relatively inactive, slowly rotating, and with no evidence of spectroscopic binarity. These are the preferred targets for follow-up spectroscopy to measure {alpha}.