The efficiency of conversion of electromagnetic energy into heat is a key factor for magnetic nanoparticles (MNPs) utilization in magnetic hyperthermia as well as in advanced magnetothermal medical techniques. Currently, the need to improve the MNPs heat release results in the increase of weight, size and power of alternating current (AC) facility and the aggravating side effects on health tissues. The aim of our work is to address these problems by offering a new strategy for improving the heat release at a relatively low frequency using ferrite based MNPs. We demonstrate nontrivial dependence of MNPs heat release on the amplitude of the AC magnetic field in zinc manganese ferrite Zn0.2Mn0.8Fe2O4 MNPs (close to 5th power instead of a well-known quadratic). Our studies challenge the current conventional approaches based on the minimization of magnetic field amplitude. We offer a better strategy: to increase the amplitude while reducing the frequency to exploit the full potential of superquadratic law for amplitude dependence of heat output. We believe that this research should have a profound impact on a number of advanced biomedical applications of MNPs and magnetic hyperthermia.