Abstract Background Graph-based reference genomes have become popular as they allow read mapping and follow-up analyses in settings where the exact haplotypes underlying a high-throughput sequencing experiment are not precisely known. Two recent papers show that mapping to graph-based reference genomes can improve accuracy as compared to methods using linear references. Both of these methods index the sequences for most paths up to a certain length in the graph in order to enable direct mapping of reads containing common variants. However, the combinatorial explosion of possible paths through nearby variants also leads to a huge search space and an increased chance of false positive alignments to highly variable regions. Results We here assess three prominent graph-based read mappers against a hybrid baseline approach that combines an initial path determination with a tuned linear read mapping method. We show, using a previously proposed benchmark, that this simple approach is able to improve overall accuracy of read-mapping to graph-based reference genomes. Conclusions Our method is implemented in a tool Two-step Graph Mapper, which is available at https://github.com/uio-bmi/two_step_graph_mapperalong with data and scripts for reproducing the experiments. Our method highlights characteristics of the current generation of graph-based read mappers and shows potential for improvement for future graph-based read mappers.