You can move your Zimbra volumes using the ZeXtras Powerstore module of ZeXtras Suite in two different ways:
Set up HSM: Using the Hierarchical Storage Management you can automatically move data from primary to secondary volumes according to a set of policies you design. This is especially useful when you have different level of storage to keep the most recent data on the fastest storage and older data on the slowest (e.g. recent data on a SAS FC storage and old data on SATA3). In your specific case, I suggest to create a Secondary volume on the new disk - setting it as both 'Compressed' and 'Current' - and set up an HSM Policy that will move any email and document older than 30 days to the secondary volume.
Decommission the old volume: using the 'doVolumeToVolume' move CLI command you can move the full content of a volume into another. You'll need to create a new Primary volume on the new disk and set it as Current in order to be able to move your data there.
Furthermore, the item deduplication feature of ZxPowerstore can further help is saving disk space.