Edit single page from plugin

You could do this with the the_content filter:

/**
 * Adds some text to bottom of the content on 'season' pages.
 *
 * @param   string  $content    The current content of the season page.
 * @return  string              The new content of the season page.
 */
function add_season_content( $content ) {

    if ( is_singular( 'season' ) && is_main_query() ) {

        // collect your season data over here...
        $season_data="Hello world";

        $content .= $season_data;
    }

    return $content;

}

add_filter( 'the_content', 'add_season_content' );

Place this code anywhere inside your plugin files.