Double perovskites are a growing class of compounds with prospects for realization of novel magnetic behaviors. The rich chemistry of double perovskites calls for high-throughput computational screening that can be followed by or combined with machine-learning techniques. Yet, most approaches neglect the bulk of microscopic information implicitly provided by first-principles calculations, severely reducing the predictive power. In this work, we remedy this drawback by including onsite energies and transfer integrals between the d states of magnetic atoms. These quantities were computed by Wannierization of the relevant energy bands. By combining them with the experimental information on the magnetism of studied materials and applying machine learning, we constructed a model capable of predicting the magnetic properties of the remaining materials whose magnetism has not been addressed experimentally. Our approach combines classification learning to distinguish between double perovskites with dominant ferromagnetic or antiferromagnetic interactions and regression employed to estimate magnetic transition temperatures. In this way, we identified one antiferromagnet and three ferromagnets with a high transition temperature. Another 28 antiferromagnetic candidates were identified as magnetically frustrated compounds. Among them, cubic Ba₂LaReO₆ shows the highest frustration parameter, which is further validated by a direct first-principles calculation. Our methodology holds promise for eliminating the need for resource-demanding calculations.