The spectral shape and flavor composition of the high-energy astrophysical neutrino flux can contain important information about the sources and processes which produce it. The IceCube Neutrino Observatory has previously demonstrated the ability to observe neutrinos of all flavors by selecting events which interact within the detector volume. Sensitivity to charged current muon neutrino interactions in or close to the detector has also been shown by selecting muon track events whose directions indicate passage through the Earth. We present an updated analysis of starting events using 6 years of IceCube data taken from 2010--2016 focusing on energies from the PeV region down to 1 TeV, far below the threshold of the original data sample used in the initial discovery of the astrophysical flux. Astrophysical neutrinos remain the dominant component in the southern sky down to 10 TeV. We then also perform a unified analysis of the flavor and spectrum implications of this sample when combined with the recently published data on $\nu_\mu$ induced muon tracks as well as recent work to identify candidate $\nu_\tau$ events.