Try adding these rules (keep the ones you already have):
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_URI} ^/([^/]+)/.+$
RewriteCond %{DOCUMENT_ROOT}/%1.php -f
RewriteRule ^([^/]+)/(.+)$ /$1.php?page=$2 [L]
This is more or less similar to what you have, with the exception of using %1backreference to check if the first part of the URI exists as a php file. This is a match for the first part:
RewriteCond %{REQUEST_URI} ^/([^/]+)/.+$
And this is fheck to see if grouping exists ([^/]+)as php file:
RewriteCond %{DOCUMENT_ROOT}/%1.php -f
, /, , page.