The emergence of CNNs in mainstream deployment has necessitated methods to design and train efficient architectures tailored to maximize the accuracy under diverse hardware and latency constraints. To scale these resource-intensive tasks with an increasing number of deployment targets, Once-For-All (OFA) proposed an approach to jointly train several models at once with a constant training cost. However, this cost remains as high as 40-50 GPU days and also suffers from a combinatorial explosion of sub-optimal model configurations. We seek to reduce this search space -- and hence the training budget -- by constraining search to models close to the accuracy-latency Pareto frontier. We incorporate insights of compound relationships between model dimensions to build CompOFA, a design space smaller by several orders of magnitude. Through experiments on ImageNet, we demonstrate that even with simple heuristics we can achieve a 2x reduction in training time and 216x speedup in model search/extraction time compared to the state of the art, without loss of Pareto optimality! We also show that this smaller design space is dense enough to support equally accurate models for a similar diversity of hardware and latency targets, while also reducing the complexity of the training and subsequent extraction algorithms. Our source code is available at https://github.com/gatech-sysml/CompOFA