Abstract: We studied the electronic and magnetic properties of iron-chalcogenide Fe$_{1+y}$Te compounds based on the multiband model, in which localized spins and itinerant electrons coexist and are coupled by Hund's rule coupling. Integrating out the conduction electrons, we computed additional couplings between localized spins similarly to the conventional Ruderman-Kittel-Kasuya-Yosida (RKKY)theory. We found that resulting RKKY-like interactions are substantial up to the third neighbors, but are negligible beyond this. We computed the magnetic phase diagram of the modified $J_1-J_2-J_3-K$ model and showed it captures the evolution of the magnetic order in Fe$_{1+y}$Te as a function of Fe excess $y$.