Right ok, so once you get the Mac back lets assume you have the SSD with a fresh install of OSX, the internal drive has all the data still on it and you also have an external drive with your most recent backup before the change.
Boot the Mac up and ensure you you have booted off the correct disc, if you are unsure go to System Preferences and check the startup disc, ( also boot time should tell you without doing this )
If you are still using the old HD choose the other disc and restart machine, once booted to SSD create a user account , ideally NOT the same as the account on your old HD, call it Admin or whatever. ( Your migration will pull in your user account so that is why I recommend using a different name, once finished you can remove this account account )
Once logged in, wipe the internal HD so you now only have the SSD as the only bootable option.
Next connect the external with your backup and go to Migration assistant , run it and point back to your External drive with the data on.
I think you may be able to actually use setup assistant and use the backup to migrate instead of creating an account, if you want some info on both see here:
https://support.apple.com/en-gb/HT204350
Hope this helps, I'm sure others will chime in too!