Reopening a session in PHP

session_start();
...
session_write_close();
...

ini_set('session.use_only_cookies', false);
ini_set('session.use_cookies', false);
ini_set('session.use_trans_sid', false);
ini_set('session.cache_limiter', null);
session_start(); // second session_start

This will prevent php from calling php_session_send_cookie() a second time.
See it working.

Though restructuring the scripts still seems to be the better option...

---

For PHP 7.2+, you will basically need to re-implement session cookies to avoid errors and warnings:

ini_set('session.use_only_cookies', false);
ini_set('session.use_cookies', false);
ini_set('session.use_trans_sid', false);
ini_set('session.cache_limiter', null);

if(array_key_exists('PHPSESSID', $_COOKIE))
    session_id($_COOKIE['PHPSESSID']);
else {
    session_start();
    setcookie('PHPSESSID', session_id());
    session_write_close();
}

session_start();
...
session_write_close();
...

session_start(); // second session_start

EDIT See @VolkerK's solution, it is better than this one.

Just buffer the output of your script while it executes to prevent the headers from being sent, and output it at the very end:

<?php

  session_start();
  // setting all the session vars I like to set
  session_write_close();

  // Start output buffer
  ob_start();

  // Code that takes some time to execute

  // Do session stuff at the end of the script
  session_start();
  $_SESSION['key'] = 'new value I like to store';
  session_write_close();

  // Send output to client
  ob_end_flush();

Ref: Output control functions