We have measured the 3He abundance from approximately 0.5 to 2 MeV nucleon-1 in 12 large solar energetic particle (SEP) events during the period 1997 November-1999 June. In five of the events, the 3He time-intensity profile is similar to the 4He time-intensity profile, indicating a common acceleration and transport origin for the two species. The average 3He/4He ratio during these events is &parl0;1.9+/-0.2&parr0;x10-3, a factor of approximately 5 enhancement over the solar wind value. During this same survey period, we have also measured the low-energy ion intensities during quieter periods in between the large-particle events. We find 3He and Fe remnants from impulsive events present on a majority of the days, implying that they fill a substantial volume (>50%) of the in-ecliptic interplanetary medium during our survey. We suggest that these suprathermal ions may therefore be a source population that is available for further acceleration by interplanetary shocks that accompany large SEP events, thereby leading to the 3He enhancements in a significant fraction of large SEP events. This impulsive SEP event material might also account for recent observations of large solar particle events with energetic particle ionization states that have a wide range of ionization states that encompass values expected for both gradual and impulsive solar SEP events.