Coarse-grained reconfigurable architectures (CGRAs) are programmable logic devices with large coarse-grained ALU-like logic blocks, and multi-bit datapath-style routing. CGRAs often have relatively restricted data routing networks, so they attract CAD mapping tools that use exact methods, such as Integer Linear Programming (ILP). However, tools that target general architectures must use large constraint systems to fully describe an architecture's flexibility, resulting in lengthy run-times. In this paper, we propose to derive connectivity information from an otherwise generic device model, and use this to create simpler ILPs, which we combine in an iterative schedule and retain most of the exactness of a fully-generic ILP approach. This new approach has a speed-up geometric mean of 5.88x when considering benchmarks that do not hit a time-limit of 7.5 hours on the fully-generic ILP, and 37.6x otherwise. This was measured using the set of benchmarks used to originally evaluate the fully-generic approach and several more benchmarks representing computation tasks, over three different CGRA architectures. All run-times of the new approach are less than 20 minutes, with 90th percentile time of 410 seconds. The proposed mapping techniques are integrated into, and evaluated using the open-source CGRA-ME architecture modelling and exploration framework.