The GMR ($d_Cu$) oscillatory behaviour as well as the widths of first and the second antiferromagnetically coupled ranges of the Permalloy ($Py=Ni_{83}Fe_{17}$)/Cu multilayers have been found to be strongly affected not only by the presence of the superparamagnetic/paramagnetic entitles located at the Py/Cu interfaces but mainly by the existence of the magnetic bridges between Py layers. The effectiveness of the magnetic bridges has been found to be temperature dependent, leading to the temperature dependence of the remnant to saturation magnetization ratio ($M_R/M_S$). We have found that for Py/Cu multilayers with equal Py and Cu layer thicknesses a high field sensitivity of the GMR effect (0.4%/Oe) and negligible hysteresis can be achieved when the number of Py layers decreases from 100 to 6. Sensitivity can be further improved by increasing the Py layers thickness, but the hysterstic effect becomes more pronounced then.