In this paper, we present compression techniques for effectively reducing the test data volume requirements of modern systems-on-chip (SoCs). Our techniques are based on the following observations: (i) conventional test compression schemes, which are designed to satisfy various constraints including low hardware over-heads and low decompression times, cannot fully exploit compression opportunities present in test data, and (ii) due to the diversity of components used in SoCs (and consequently in their test strategies and test data characteristics), a single compression strategy may not be best suited to handle them. We propose the use of multi-level and heterogeneous test compression schemes to address the above issues, and demonstrate that they can provide significant reductions in test volume above currently known state-of-the-art test compression techniques. We also suggest various architectural customization techniques such as partitioning of decompression functionality between hardware and software, and addition of custom instructions, to improve decompression times and reduce hardware overheads. Experiments with several designs, including an industrial media processing SoC, demonstrate the efficacy of the proposed techniques in achieving test data volume reductions with low overheads.