How to add a second content section when using certain page template

You could add a metabox for your second content section, then hide it with a bit of javascript if the current template isn’t a specific template:

current_template = jQuery('#page_template').val();
if( current_template != 'my-special-template.php' ){
    jQuery('#_your_meta_box_id').hide();
}

then bind a function to the change event of the template dropdown and show it when a specific template is selected:

jQuery('#page_template').change(function(){
    selected_template = jQuery(this).val();
    if( selected_template == 'my-special-template.php' ){
        jQuery('#_your_meta_box_id').show();
    }
});