We performed magnetic-field-induced experiments on micron-sized patterned Co70.4Fe4.6Si15B10 square thin-film elements with in-plane magnetic anisotropy by magneto-optical scanning near-field optical microscopy (MO-SNOM) with a spatial resolution better than 200 nm. Markedly different local hysteresis loops (LHLs) were measured on selected positions in one element. Some LHLs presented an unusual shape intrinsic of local magnetic-field-induced process. Comparison of the MO-SNOM imaging results with high-resolution far-field Kerr microscopy has confirmed the local character of the MO-SNOM measurements. This has also helped us to understand the unusual LHLs shapes as related to the field-induced rearrangement of the domain structure within the square element during the magnetization process. The magnetic structure in small field is well described by two overlapping four-domain flux-closure configurations that are well modeled by micromagnetic calculations.