Nowadays, the rapid growth of Deep Neural Network (DNN) architectures has established them as the defacto approach for providing advanced Machine Learning tasks with excellent accuracy. Targeting low-power DNN computing, this paper examines the interplay of fine-grained error resilience of DNN workloads in collaboration with hardware approximation techniques, to achieve higher levels of energy efficiency. Utilizing the state-of-the-art ROUP approximate multipliers, we systematically explore their fine-grained distribution across the network according to our layer-, filter-, and kernel-level approaches, and examine their impact on accuracy and energy. We use the ResNet-8 model on the CIFAR-10 dataset to evaluate our approximations. The proposed solution delivers up to 54% energy gains in exchange for up to 4% accuracy loss, compared to the baseline quantized model, while it provides 2x energy gains with better accuracy versus the state-of-the-art DNN approximations.