Yes, you can manage this with Partition Wizard (or Partition Magic) by resizing (compressing) the storage disk partition and adding a partition. But, when you do this you should clone the 128 GB SSD and place that on the HDD. When you install the new 256 GB SSD you will have to do a clean install on the drive, with the HDD disconnected (otherwise the system will see the old OS and boot from it). After installation of the SSD you will reconnect the HDD and wipe out the cloned 128 GB partition and stretch the HDD back to its original size.
If I were you I wouldn't do this but, would create a restore file of the C: drive on the HDD, and wait for the delivery of the new SSD. When the new SSD came I'd use the restore function of Windows to recreate the original OS disk on the 256 GB SSD. If need be, I'd use Partition Wizard to stretch the 256 SSd to its full capacity. It is simpler, safer, and you will be out of a computer for a few days. But you have to choose your own poison.