There is increasing interest in distilling task-specific knowledge from large language models (LLM) to smaller student models. Nonetheless, LLM distillation presents a dual challenge: 1) there is a high cost associated with querying the teacher LLM, such as GPT-4, for gathering an ample number of demonstrations; 2) the teacher LLM might provide imperfect outputs with a negative impact on the student's learning process. To enhance sample efficiency within resource-constrained, imperfect teacher scenarios, we propose a three-component framework leveraging three signal types. The first signal is the student's self-consistency (consistency of student multiple outputs), which is a proxy of the student's confidence. Specifically, we introduce a ''teaching assistant'' (TA) model to assess the uncertainty of both the student's and the teacher's outputs via confidence scoring, which serves as another two signals for student training. Furthermore, we propose a two-stage training schema to first warm up the student with a small proportion of data to better utilize student's signal. Experiments have shown the superiority of our proposed framework for four complex reasoning tasks. On average, our proposed two-stage framework brings a relative improvement of up to 20.79% compared to fine-tuning without any signals across datasets.