We examined solar energetic particle (SEP) events from May 1996 through April 2000 that showed evidence for two peaks of intensity, separated by 3--24 hours, in the energy range ~1--20 MeV, observed with the Energetic and Relativistic Nuclei and Electron instrument (ERNE) onboard Solar and Heliospheric Observatory (SOHO). Of 88 clear SEP events observed, 18 exhibited such double peaks, with more than factor of two dips between the peaks in the energy range up to at least 8 MeV. Second peaks were examined in terms of velocity dispersion and change of 4He/p abundance ratio. Three events showed both velocity dispersion and abundance change, four events just an abundance change, and four showed velocity dispersion and no abundance change. We discuss how these different properties may reflect acceleration processes near the Sun and in the interplanetary medium. In the remaining seven events with neither velocity dispersion nor abundance change, we suggest that the second peak was a spatial feature caused by interplanetary magnetic field structures.