Over the last few years coronagraphic and spectroscopic observations have demonstrated that small-scale eruptions, such as "jets," "narrow coronal mass ejections (CMEs)," "mini CMEs," "streamer puffs," "streamer detachments," and others, occur ubiquitously on the Sun. Nevertheless, the origin of small-scale eruptive events and how these are interrelated with larger scale CMEs have been poorly investigated so far. In this work, we study a series of small-scale side eruptions that occurred during and after a large-scale CME. Observations show that a CME can be associated not only with a single reconnection process, leading to the large-scale phenomenon, but also with many other side reconnections occurring at different locations and times around the main flux rope, possibly induced by the CME expansion in the surrounding corona. White light and EUV observations of a slow CME acquired by the SOHO/LASCO and SOHO/UVCS instruments are analyzed here to characterize the locations of side reconnections induced by the CME. The magnetic reconnection rate M has been estimated from the UVCS data from the ratio between the inflows and outflows observed around the reconnection region, and from the LASCO data from the observed aperture angles between the slow mode shocks (SMSs) associated with the reconnection. It turns out that M 0.05 at the heliocentric distance of 1.8 R , while between ~2.5 and 5.5 R , M values progressively decrease with time/altitude from M ~ 1 down to M ~ 0.3. Such large values of M are theoretically acceptable only if flux pile-up reconnection is envisaged. The observed occurrence of multiple reconnections associated with a CME is verified by numerical simulations of an eruption occurring within multiple helmet streamers. The simulations confirm that small side reconnections are a consequence of CME expansion against the surrounding coronal streamers. The simulated and observed evolution of aperture angles between the SMSs are in good agreement as well. These results demonstrate the effect of the global coronal magnetic field in the occurrence of small-scale eruptions due to lateral reconnection in a preceding CME event.