The effect of flux trapping on the flux-voltage characteristics of multi-loop SQUID magnetometers was investigated by means of repeated cool-down cycles in a stepwise increased magnetic background field. For a SQUID with N parallel loops, N different flux-offsets, each separated by Φ_0/N, were observed even in zero magnetic field. These flux offsets further split into a so called fine structure, which can be explained by minor asymmetries in the SQUID design. The observed results are discussed with peculiar regard to their impact on the previously presented absolute SQUID cascade vector magnetometer.