Interstellar neutral (ISN) hydrogen is the most abundant species in the outer heliosheath and the very local interstellar medium (VLISM). Charge-exchange collisions in the outer heliosheath result in filtration, reducing the ISN hydrogen density inside the heliosphere. Additionally, these atoms are intensively ionized close to the Sun, resulting in a substantial reduction of their density within a few astronomical units from the Sun. The products of this ionization—pickup ions (PUIs)—are detected by charged particle detectors. The Solar Wind Around Pluto instrument on New Horizons provides, for the first time, PUI observations from the distant heliosphere. We analyze the observations collected between 22 and 52 au from the Sun to find the ISN hydrogen density profile and compare the results with predictions from global heliosphere models. We conclude that the density profile derived from the observations is inconsistent with steady-state model predictions. This discrepancy is not explained by time variations close to the Sun and thus may be related to the temporal evolution of the outer boundaries or VLISM conditions. Furthermore, we show that the cold and hot models of ISN hydrogen distribution are not a good approximation closer to the termination shock. Therefore, we recommend a new fiduciary point based on the available New Horizons observations at 40 au from the Sun, at ecliptic direction (285.°62, 1.°94), where the ISN hydrogen density is 0.11 cm−3. The continued operation of New Horizons should give better insight into the source of the discussed discrepancy.