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This study provides a detailed analysis of five distant interplanetary shocks observed by the Solar Wind Around Pluto instrument on board New Horizons, which exhibit the signature of a suprathermal H ^+ pickup ion (PUI) tail in the downstream distribution. These shocks were observed with a PUI data cadence of approximately 24 hr, covering a heliocentric distance range of 23.71–36.75 au. The shock compression ratio varies between approximately 1.4 and 3.2. The H ^+ PUI density and temperature show a gradual increase across the shock, while the H ^+ solar wind density shows erratic behavior without a distinct downstream compression. The H ^+ PUI cooling index variation across the shock displays different characteristics in each shock. This study demonstrates, for the first time, the variation of the number density of downstream H ^+ PUI tails with the shock compression ratio, revealing an increase in tail density with stronger shocks. Additionally, theoretical estimates of reflected PUI number densities derived from the electrostatic cross-shock potential agree very well with the observed H ^+ PUI tail densities for stronger shocks.