I'm not sure what you just did there, but from what I can tell this is what you're asking for:
bookingfacilities.php
<form action="successfulbooking.php" method="post">
<input type="hidden" name="date" value="<?php echo $date; ?>">
<input type="submit" value="Submit Form">
</form>
successfulbooking.php
<?php
$date = $_POST['date'];
// add code here
?>
Not sure what you want to do with that third page(booking_now.php) too.