We report on the direct search for cosmic relic neutrinos using data acquired during the first two science campaigns of the KATRIN experiment in 2019. Beta-decay electrons from a high-purity molecular tritium gas source are analyzed by a high-resolution MAC-E filter around the endpoint at 18.57 keV. The analysis is sensitive to a local relic neutrino overdensity ratio of η < 9.7 · 1010/α (1.1 · 1011/α) at a 90% (95%) confidence level with α = 1 (0.5) for Majorana (Dirac) neutrinos. A fit of the integrated electron spectrum over a narrow interval around the endpoint accounting for relic neutrino captures in the tritium source reveals no significant overdensity. This work improves the results obtained by the previous neutrino mass experiments at Los Alamos and Troitsk. We furthermore update the projected final sensitivity of the KATRIN experiment to η < 1 · 1010/α at 90% confidence level, by relying on updated operational conditions.