Abstract: High-energy muon and electron neutrinos yield a non-negligible flux of tau neutrinos as they propagate through Earth. In this Letter, we address the impact of this additional component in the PeV and EeV energy regimes for the first time. Above 300 TeV, this contribution is predicted to be significantly larger than the atmospheric background, and it alters current and future neutrino telescopes' capabilities to discover a cosmic tau-neutrino flux. Further, we demonstrate that Earth-skimming neutrino experiments, designed to observe tau neutrinos, will be sensitive to cosmogenic neutrinos even in extreme scenarios without a primary tau-neutrino component.