Uses the WC session manager to ensure checkout details are not lost after a customer returns after leaving the checkout page.

woocommerce-persistent-checkout-fields.php

<?php
add_action( 'woocommerce_after_checkout_validation', 'set_persitent_checkout' );
add_filter( 'woocommerce_checkout_get_value', 'get_persistent_checkout', 10, 2 );
add_action( 'woocommerce_ship_to_different_address_checked', 'get_persitent_ship_to_different' );

function set_persitent_checkout ( $a ) 
{
    $arr = array();
    foreach ( $a as $key => $value )
        if ( ! empty($value) )
            $arr[$key] = $value;
            
    WC()->session->set( 'form_data', $arr );
    return $a;
}

function get_persistent_checkout ( $value, $index ) 
{
    $data = WC()->session->get('form_data');
    if ( ! $data || empty($data[$index]) )
        return $value;
    return is_bool($data[$index]) ? (int) $data[$index] : $data[$index];
}

function get_persitent_ship_to_different ( $value ) 
{
    $data = WC()->session->get('form_data');
    if ( ! $data || empty($data['ship_to_different_address']) )
        return $value;
        
    return is_bool($data['ship_to_different_address']) ? (int) $data['ship_to_different_address'] : $data['ship_to_different_address'];
}