On your old phone, make sure you're signed into your Samsung account, go to Settings, and scroll down to accounts and backup. Click on the backup that has "Samsung cloud" as the subheading, then toggle the "Messages" on and click on "Back up now." After the backup is complete, go to your new phone, also making sure you're signed into your Samsung account, go to Settings>Accounts and backup>Restore data (with Samsung cloud as the subheading) then click on the old phone from the list of options that show up, check-mark "Messages" and then tap on "Restore" after which you can check your messages on your new phone to see if it transferred.
@Pointmasteriam You can restart Smart Switch. Select Custom when it allows you to choose which type of transfer. Turn off everything except your messages. Proceed with the transfer and it will only transfer your messages.
