Short-read aligners predominantly use the FM-index, which is easily able to index one or a few human genomes. However, it does not scale well to indexing collections of thousands of genomes. Driving this issue are the two chief components of the index: (1) a rank data structure over the Burrows–Wheeler Transform (BWT) of the string that will allow us to find the interval in the string’s suffix array (SA), and (2) a sample of the SA that—when used with the rank data structure—allows us to access the SA. The rank data structure can be kept small even for large genomic databases, by run-length compressing the BWT, but until recently there was no means known to keep the SA sample small without greatly slowing down access to the SA. Now that (SODA 2018) has defined an SA sample that takes about the same space as the run-length compressed BWT, we have the design for efficient FM-indexes of genomic databases but are faced with the problem of building them. In 2018, we showed how to build the BWT of large genomic databases efficiently (WABI 2018), but the problem of building the sample efficiently was left open. We compare our approach to state-of-the-art methods for constructing the SA sample, and demonstrate that it is the fastest and most space-efficient method on highly repetitive genomic databases. Lastly, we apply our method for indexing partial and whole human genomes and show that it improves over the FM-index-based Bowtie method with respect to both memory and time and over the hybrid index-based CHIC method with respect to query time and memory required for indexing.