A technique forms contents of a compression container from a standalone container file system stored on the same data storage array on which the compression container is stored. Such a standalone container includes a container file which is formed by standalone blocks. The standalone blocks store host data and which are hierarchically related to each other. Upon receiving a compression command, the storage processor carries out a set of compression operations which include creating and storing the compression container in the data storage array. The compression container includes container files which are formed by compression blocks storing the host data and which are hierarchically related to each other.