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While unobscured and radio-quiet active galactic nuclei are regularly being found at redshifts
$z > 6$
, their obscured and radio-loud counterparts remain elusive. We build upon our successful pilot study, presenting a new sample of low-frequency-selected candidate high-redshift radio galaxies (HzRGs) over a sky area 20 times larger. We have refined our selection technique, in which we select sources with curved radio spectra between 72–231 MHz from the GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) survey. In combination with the requirements that our GLEAM-selected HzRG candidates have compact radio morphologies and be undetected in near-infrared
$K_{\rm s}$
-band imaging from the Visible and Infrared Survey Telescope for Astronomy Kilo-degree Infrared Galaxy (VIKING) survey, we find 51 new candidate HzRGs over a sky area of approximately
$1200\ \mathrm{deg}^2$
. Our sample also includes two sources from the pilot study: the second-most distant radio galaxy currently known, at
$z=5.55$
, with another source potentially at
$z \sim 8$
. We present our refined selection technique and analyse the properties of the sample. We model the broadband radio spectra between 74 MHz and 9 GHz by supplementing the GLEAM data with both publicly available data and new observations from the Australia Telescope Compact Array at 5.5 and 9 GHz. In addition, deep
$K_{\rm s}$
-band imaging from the High-Acuity Widefield K-band Imager (HAWK-I) on the Very Large Telescope and from the Southern Herschel Astrophysical Terahertz Large Area Survey Regions
$K_{\rm s}$
-band Survey (SHARKS) is presented for five sources. We discuss the prospects of finding very distant radio galaxies in our sample, potentially within the epoch of reionisation at
$z \gtrsim 6.5$
.
The GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) is a radio continuum survey at 76–227 MHz of the entire southern sky (Declination 
$<\!{+}30^{\circ}$) with an angular resolution of 
${\approx}2$ arcmin. In this paper, we combine GLEAM data with optical spectroscopy from the 6dF Galaxy Survey to construct a sample of 1 590 local (median 
$z \approx 0.064$) radio sources with 
$S_{200\,\mathrm{MHz}} > 55$ mJy across an area of 
${\approx}16\,700\,\mathrm{deg}^{2}$. From the optical spectra, we identify the dominant physical process responsible for the radio emission from each galaxy: 73% are fuelled by an active galactic nucleus (AGN) and 27% by star formation. We present the local radio luminosity function for AGN and star-forming (SF) galaxies at 200 MHz and characterise the typical radio spectra of these two populations between 76 MHz and 
${\sim}1$ GHz. For the AGN, the median spectral index between 200 MHz and 
${\sim}1$ GHz, 
$\alpha_{\mathrm{high}}$, is 
$-0.600 \pm 0.010$ (where 
$S \propto \nu^{\alpha}$) and the median spectral index within the GLEAM band, 
$\alpha_{\mathrm{low}}$, is 
$-0.704 \pm 0.011$. For the SF galaxies, the median value of 
$\alpha_{\mathrm{high}}$ is 
$-0.650 \pm 0.010$ and the median value of 
$\alpha_{\mathrm{low}}$ is 
$-0.596 \pm 0.015$. Among the AGN population, flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of weak radio AGN that remain core-dominated even at low frequencies. However, around 4% of local radio AGN have ultra-steep radio spectra at low frequencies (
$\alpha_{\mathrm{low}} < -1.2$). These ultra-steep-spectrum sources span a wide range in radio luminosity, and further work is needed to clarify their nature.
