It does create a new one for you, as long as the new directory has the correct permissions. If you have made any modifications to your .htaccess file you should take a backup of it first so you can re-apply those changes to the newly generated file.
If you have SSL enabled login to the admin side of the store. Go to "Settings" then the SSL tab. Update the paths to reflect the new one. Once done create the folder you want to move it to. Select all the files with an FTP client and move them to the new folder. Empty the stores cache folder. All should be fine.