Assuming you have a complete Time Machine backup on your external drive, add another partition on the drive, namely a ~10 GB one, formatted as HFS+.
From Disk Utility, select this partition, and click the Restore tab to restore the Snow Leopard installer disk image onto this partition. Select the Snow Leopard image and the destination partition by dragging the partition from the left pane to the Destination field.
After that, swap your internal hard drives and boot up with the external drive. In order to do this, hold ⌥ while booting. You'll land in the Snow Leopard installer if the image has been restored correctly.
Here, after clicking Install Mac OS X (or similar), select the Utilities menu from the top, and from there, Restore from backup. Select your existing Time Machine backup and let it restore the backup to your SSD.
You could also do a clean installation of Snow Leopard on the SSD before – completely neglecting your Time Machine backup. Later, use the Migration Assistant from /Applications/Utilities to restore only needed data from the Time Machine backup.