Scientific simulations on high-performance computing systems produce vast amounts of data that need to be stored and analyzed efficiently. Lossy compression significantly reduces the data volume by trading accuracy for performance. Despite the recent success of lossy compression, such as ZFP and SZ, the compression performance is still far from being able to keep up with the exponential growth of data. This paper aims to further take advantage of application characteristics, an area that is often under-explored, to improve the compression ratios of adaptive mesh refinement (AMR) - a widely used numerical solver that allows for an improved resolution in limited regions. We propose a level reordering technique zMesh to reduce the storage footprint of AMR applications. In particular, we group the data points that are mapped to the same or adjacent geometric coordinates such that the dataset is smoother and more compressible. Unlike the prior work where the compression performance is affected by the overhead of metadata, this work re-generates restore recipe using a chained tree structure, thus involving no extra storage overhead for compressed data, which substantially improves the compression ratios. The results demonstrate that zMesh can improve the smoothness of data by 67.9% and 71.3% for Z-ordering and Hilbert, respectively. Overall, zMesh improves the compression ratios by up to 16.5% and 133.7% for ZFP and SZ, respectively. Despite that zMesh involves additional compute overhead for tree and restore recipe construction, we show that the cost can be amortized as the number of quantities to be compressed increases.