A technique manages host data in a data storage array. The technique involves placing, in response to host input/output (I/O) requests from a set of host devices, host data in a container file system which is stored in a slice pool of storage slices (e.g., storage space of a standard size such as 256 MB) formed by multiple storage tiers of the data storage array. Each storage tier provides storage access at a different storage access speed. The technique further involves assigning classifications to storage slices of the slice pool which store the container file system, each classification being assigned based on storage slice access history resulting from the host I/O requests. The technique further involves relocating portions of the container file system among the multiple storage tiers according to the classifications assigned to the storage slices of the slice pool.