Moving SQL Server database to a new disk while online
One method to move the entire database is with BACKUP and RESTORE.  The database will be unavailable during the final switch but downtime should be minimal with planning.  This procedure assumes the FULL or BULK_LOGGED recovery model:

1) Perform a FULL backup (or use your existing one).

2) Restore latest full backup to a different database name, specifying the WITH MOVE option to relocate files on SSD storage as desired and the NORECOVERY option so that subsequent differential and log backups can be restored.

3) Apply incremental changes to the new database until the time of the final cut-over with transaction log backups and RESTORE...WITH NORECOVERY.  This will minimize downtime for the final switch to the new database.

4) To switch to the new database, take the application offline, perform a final transaction log backup, and apply to the new database WITH RECOVERY.  Finally, rename the original database to a different name and rename the relocated database to the original name.  Drop the old database at your convenience.

In the SIMPLE recovery model, you could use a similar process but without the step 3 transaction log backup/restores.  Instead, use a differential database backup/restore in the final step.  That may require more downtime, depending on the amount of changes since the initial FULL backup.