Is there a hook or filter that adds a button to the left of the search box?

After reviewing the WordPress core code that renders the search box:

/**
 * Displays the search box.
 *
 * @since 4.6.0
 *
 * @param string $text     The 'submit' button label.
 * @param string $input_id ID attribute value for the search input field.
 */
public function search_box( $text, $input_id ) {
    if ( empty( $_REQUEST['s'] ) && ! $this->has_items() ) {
        return;
    }

    $input_id = $input_id . '-search-input';

    if ( ! empty( $_REQUEST['orderby'] ) ) {
        echo '<input type="hidden" name="orderby" value="' . esc_attr( $_REQUEST['orderby'] ) . '" />';
    }
    if ( ! empty( $_REQUEST['order'] ) ) {
        echo '<input type="hidden" name="order" value="' . esc_attr( $_REQUEST['order'] ) . '" />';
    }
    ?>
    <p class="search-box">
        <label class="screen-reader-text" for="<?php echo esc_attr( $input_id ); ?>"><?php echo $text; ?>:</label>
        <input type="search" id="<?php echo esc_attr( $input_id ); ?>" class="wp-filter-search" name="s" value="<?php _admin_search_query(); ?>" placeholder="<?php esc_attr_e( 'Search installed plugins...' ); ?>"/>
        <?php submit_button( $text, 'hide-if-js', '', false, array( 'id' => 'search-submit' ) ); ?>
    </p>
    <?php
}

It doesn’t look like there is a hook. You’ll have to inject the button using JavaScript.