Innovations in genome sequencing techniques are enabling remarkably fast and low cost production of raw genome data. As Moore's law tapers off, bottlenecks in genome sequencing are shifting to computational resources for mapping reads to reference DNA. This paper presents SeedEx, a read-alignment accelerator focused on the seed-extension step. SeedEx is based on the observation that only a small fraction of reads require large edit distance for alignment, hence an area efficient narrow-band seed-extension accelerator can suffice in practice. However, due to the highly error-sensitive nature of genomic workloads, guaranteeing optimality of alignment result is of cardinal importance. Towards this end, we propose a speculation-and-test based framework by using strict but powerful optimality checking mechanisms. We demonstrate SeedEx by an implementation on a cloud FPGA. SeedEx achieves 6.0× iso-area throughput speedup when compared to a banded Smith-Waterman baseline, and achieving 43.9 M seed extentions/s on AWS f1.2xlarge instance. Integration with BWA-MEM2 improves the execution time by 2.3×.