A new segment of data is copied to a volatile, primary cache based on a host data read access request. The primary cache mirrors a first portion of a non-volatile main storage criterion is determined for movement of data from the primary cache to a non-volatile, secondary cache that mirrors a second portion of the main storage. The criterion gives higher priority to segments having addresses not yet selected for reading by the host. In response to the new segment of data being copied to the primary cache, a selected segment of data is copied from the primary cache to the secondary cache in response to the selected segment satisfying the criterion.