We report new observations by the Solar Wind Around Pluto (SWAP) instrument on the New Horizons spacecraft of an energy-per-charge (E/q) spectrum of interstellar pick-up ions (PUIs) from an unprecedented heliocentric distance of 17 AU. This E/q spectrum is fit well by an isotropic PUI distribution function combined with the detailed response of the SWAP instrument. In contrast to earlier work, we are also able to fit an isotropic PUI model to an E/q spectrum measured by SWAP at 11.3 AU by explicitly including two additional effects. These are (1) the E/q-dependent geometric factor of SWAP, which increases with decreasing E/q owing to effects associated with the post-acceleration of particles exiting the electrostatic analyzer portion of the instrument; and (2) a solar wind distribution, the model spectrum of which contributes significantly to the low-E/q part of the overall model owing, presumably, to secondary particles produced within the instrument.