We report on magnetotransport experiments performed at 4.2 K on hybrid structures consisting of a metal and a mesoscopic two-dimensional electron system in an InAs/InGaAs heterostructure. The devices were fabricated using cleaved-edge overgrowth. We find that they exhibit an extraordinary magnetoresistance effect (EMR) which is most pronounced in the case of the lowest specific contact resistance ρ i of ≈10−8 Ω cm2 achieved in this work. The largest relative resistance change Δ R/R is 115,000% at a magnetic field B = 1 T. A systematic study of the performance of the EMR devices with down to sub-μm lateral dimension and with different ρ i is reported.