In the IceCube Neutrino Observatory, a signal of astrophysical neutrinos is obscured by backgrounds from atmospheric neutrinos and muons produced in cosmic-ray interactions. IceCube event selections used to isolate the astrophysical neutrino signal often focus on the morphology of the light patterns recorded by the detector. The analyses presented here use the new IceCube Enhanced Starting Track Event Selection (ESTES), which identifies events likely generated by muon–neutrino interactions within the detector geometry, focusing on neutrino energies of 1–500 TeV with a median angular resolution of 1.4. Selecting for starting-track events filters out not only the atmospheric-muon background but also the atmospheric-neutrino background in the southern sky. This improves IceCube’s muon–neutrino sensitivity to southern-sky neutrino sources, especially for Galactic sources that are not expected to produce a substantial flux of neutrinos above 100 TeV. In this work, the ESTES sample was applied for the first time to search for astrophysical sources of neutrinos, including a search for diffuse neutrino emission from the Galactic plane. No significant excesses were identified from any of the analyses; however, constraining limits are set on the hadronic emission from TeV gamma-ray Galactic plane objects and models of the diffuse Galactic plane neutrino flux.