The short answer is, YES. You do have to save any fields in a session that you intend to enter in the orders table.
Around line 222 on the process_transaction.php page you have to add the sessions in an area marked "Store the submitted form in the session".
Then you have to add hidden input fields in the form(s) further down that same page. I added them to the "Other gateways" form in all three sections and it worked. I did not add hidden fields to the individual gateway forms even though I am using PayPal Express as the gateway.
NOTE: I am using a bastardized checkout system in the admin section of my store instance for internal clients and bypass the payment process. Otherwise I guess you might need to add hidden input fields to the PayPal gateway form as well (or SagePay, Authorize.net, etc)