Improve month view search so it is not restricted to the current month. Experimental! Use at own risk, there are known bugs/quirks with this code.

experimental-smart-month-search.php

<?php
/**
 * If a search is conducted in month view but the relevant events
 * are in a different month, switch to that month.
 *
 * Experimental! Especially during live ajax navigation, there are
 * parts of this that don't work cleanly (pushstate/URL changes).
 * Unsupported, use at own risk.
 */
class Transmogrified_Month_View_Search {
	private $month  = '';
	private $search = '';
	private $target = '';

	public function __construct() {
		$this->month = empty( $_REQUEST['tribe-bar-date'] )
			? date_create()->format( 'Y-m' )
			: $_REQUEST['tribe-bar-date'];

		$this->search = empty( $_REQUEST['tribe-bar-search'] )
			? ''
			: $_REQUEST['tribe-bar-search'];

		add_action( 'admin_init', [ $this, 'listen_for_ajax_loads' ], 1 );
		add_action( 'parse_query', [ $this, 'listen_for_direct_loads' ] );
	}

	public function listen_for_ajax_loads() {
		if ( ! defined( 'DOING_AJAX' ) || ! DOING_AJAX ) {
			return;
		}

		if ( 'tribe_calendar' !== @$_POST['action'] ) {
			return;
		}

		remove_action( 'parse_query', [ $this, 'listen_for_direct_loads' ] );
		add_filter( 'tribe_events_header_attributes', [ $this, 'month_view_header_attrs' ] );
		$this->assess();
	}

	public function listen_for_direct_loads( WP_Query $query ) {
		remove_action( 'parse_query', [ $this, 'listen' ] );

		if ( ! empty( $query->get( 'eventDate' ) ) ) {
			$this->month = $query->get( 'eventDate' );
		}

		if (
			! empty( $this->search )
			&& 'month' === $query->get( 'eventDisplay' )
		) {
			$this->assess();
		}
	}

	private function assess() {
		$events = tribe_get_events( [
			'eventDisplay'   => 'custom',
			'start_date'     => $this->month,
			's'              => $this->search,
			'posts_per_page' => 1
		] );

		if ( empty( $events ) ) {
			return;
		}

		$first_event = current( $events );
		$first_event_month = date_create( $first_event->EventStartDate )->format( 'Y-m' );

		if ( $first_event_month !== $this->month ) {
			$this->target = $first_event_month;
			$this->switch_month();
		}
	}

	private function switch_month() {
		defined( 'DOING_AJAX' ) && DOING_AJAX
			? $this->switch_for_ajax()
			: $this->switch_for_direct_loads();
	}

	private function switch_for_ajax() {
		# Naughty but functional trick
		foreach ( ['eventDate', 'tribe-bar-date'] as $key ) {
			$_POST[ $key ] = $_REQUEST[ $key ] = $this->target;
		}
	}

	private function switch_for_direct_loads() {
		wp_redirect( $this->get_target_url() );
		exit;
	}

	public function month_view_header_attrs( $attrs ) {
		$attrs['data-baseurl'] = $this->get_target_url();
		return $attrs;
	}

	private function get_target_url() {
		$url = Tribe__Events__Main::instance()->getLink( 'month', $this->target );
		return add_query_arg( 'tribe-bar-search', $this->search, $url );
	}
}

new Transmogrified_Month_View_Search;