Determining the age of the geomagnetic field is of paramount importance for understanding the evolution of the planet because the field shields the atmosphere from erosion by the solar wind. The absence or presence of the geomagnetic field also provides a unique gauge of early core conditions. Evidence for a geomagnetic field 4.2 billion-year (Gy) old, just a few hundred million years after the lunar-forming giant impact, has come from paleomagnetic analyses of zircons of the Jack Hills (Western Australia). Herein, we provide new paleomagnetic and electron microscope analyses that attest to the presence of a primary magnetic remanence carried by magnetite in these zircons and new geochemical data indicating that select Hadean zircons have escaped magnetic resetting since their formation. New paleointensity and Pb-Pb radiometric age data from additional zircons meeting robust selection criteria provide further evidence for the fidelity of the magnetic record and suggest a period of high geomagnetic field strength at 4.1 to 4.0 billion years ago (Ga) that may represent efficient convection related to chemical precipitation in Earth's Hadean liquid iron core.