  Suprathermal ions in the corona are thought to serve as seed particles for
large gradual solar energetic particle (SEP) events associated with fast and
wide coronal mass ejections (CMEs). A better understanding of the role of
suprathermal particles as seed populations for SEP events can be made by using
observations close to the Sun. We study a series of SEP events observed by the
Integrated Science Investigation of the Sun (IS$\odot$IS) suite on board the
Parker Solar Probe (PSP) from 2020 May 27 to June 2, during which PSP was at
heliocentric distances between $\sim$0.4 and $\sim$0.2 au. These events were
also observed by the Ahead Solar TErrestrial RElations Observatory (STEREO-A)
near 1 au, but the particle intensity magnitude was much lower than that at
PSP. We find that the SEPs should have spread in space as their source regions
were distant from the nominal magnetic footpoints of both spacecraft, and the
parent CMEs were slow and narrow. We study the decay phase of the SEP events in
the $\sim$1-2 MeV proton energy range at PSP and STEREO-A, and discuss their
properties in terms of both continuous injections by successive solar eruptions
and the distances where the measurements were made. This study indicates that
seed particles can be continuously generated by eruptions associated with slow
and narrow CMEs, spread over a large part of the inner heliosphere, and remain
there for tens of hours, even if minimal particle intensity enhancements were
measured near 1 au.
