Hey friend, is correct:
RewriteEngine On
RewriteBase /
RewriteCond %{REQUEST_URI} !^/yournewfolder [NC]
RewriteRule ^ /yournewfolder%{REQUEST_URI} [R=301,L]
Only this in your .htacces that is within your public_html folder (root of site)
The new folder is the folder that contains the files of your website. If you use wordpress, use Softaculous to clone the entire content of your site to the new folder.
Remember that you will have to change the information for your database. - Log into your phpMyAdmin account, export your database to a location on your computer (use the option Custom - display all possible options to export and maintain a backup of your database exported - if you miss something, you will have this insurance file to import it again).
Then open your yourdatabasename.sql file on your php editor, use the replace option and make the following change.:
EX: yoursite.com
(Replacing option to) yoursite.com/yournewfolder
save the edited file. Now go back to your phpMyAdmin account and delete your old database (Selecting all tables and using the delete option), after deleting, go to the Import option and import your new database already edited.
Have a great day.