Machine learning promises to accelerate the material discovery by enabling high-throughput prediction of desirable macro-properties from atomic-level descriptors or structures. However, the limited data available about precise values of these properties have been a barrier, leading to predictive models with limited precision or the ability to generalize. This is particularly true of lattice thermal conductivity (LTC): existing datasets of precise (ab initio, DFT-based) computed values are limited to a few dozen materials with little variability. Based on such datasets, we study the impact of transfer learning on both the precision and generalizability of a deep learning model (ParAIsite). We start from an existing model (MEGNet~\cite{Chen2019}) and show that improvements are obtained by fine-tuning a pre-trained version on different tasks. Interestingly, we also show that a much greater improvement is obtained when first fine-tuning it on a large datasets of low-quality approximations of LTC (based on the AGL model) and then applying a second phase of fine-tuning with our high-quality, smaller-scale datasets. The promising results obtained pave the way not only towards a greater ability to explore large databases in search of low thermal conductivity materials but also to methods enabling increasingly precise predictions in areas where quality data are rare.