We report on new high-sensitivity measurements from the WIND spacecraft of the spatial distributions of 30 keV/amu to 10 MeV/amu ions from corotating interaction regions (CIRs) that extend far beyond the confines of the parent high-speed solar-wind stream. Not only do late-phase MeV ions persist far into the declining solar wind, but they also show a continual gain in energy, even after sector boundary passage, until the next small increase in solar wind speed occurs. These ions are accelerated in the distant heliosphere as the reverse shock from the CIR propagates completely across the rarefaction region produced by the declining solar wind, growing in acceleration efficiency as it propagates. Energetic ions from a single CIR event are seen for a period of 17 days and ∼225° in solar longitude. The observed energy spectra can be fit to the theory of Fisk and Lee  only if shock compression increases with time so that the spectra harden significantly.