Should I change wp-config for SSL?

You better use a .htaccess file, where you force https with:

RewriteEngine On
RewriteCond %{HTTPS} off
RewriteRule ^(.*)$ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]

The file goes in the root folder of your WP installation. If there is any, put this code on the beginning of the existing file.

And you can then change the settings in the WP backend Settings, if neccessary.