Zero- to ultralow-field nuclear magnetic resonance (ZULF NMR) is a rapidly developing form of spectroscopy that drastically reduces the size and expense of portable devices with NMR capabilities. However, signal acquisition still requires a mechanism for orienting nuclear spins (e.g., generating a bulk magnetic moment for detection), and the currently employed methods only apply to a limited pool of chemicals or come at prohibitively high cost. Here, we demonstrate that the parahydrogen-based SABRE-relay method (SABRE = Signal Amplification by Reversible Exchange) can be used as a more general means of generating hyperpolarized analytes for ZULF NMR. This method is applicable to a wide range of small molecules possessing exchangeable protons, as we demonstrate here by observing zero-field J-spectra of [13C]-methanol, [1-13C]-ethanol, and [2-13C]-ethanol. We also explore the magnetic-field dependence of the proton hyperpolarization efficiency in SABRE-relay, and show the existence of a second, previously unexplored maximum at 19 mT. We further demonstrate that water does not significantly diminish SABRE-relay performance using benzylamine as polarization-transfer agent and use this to hyperpolarize ethanol extracted from a store-bought sample of vodka (1H polarization of ~0.1%). Applications for detecting trace chemical impurities and measuring J-spectra from natural extracts are also discussed.