How can I add a field I can fill after the user that will update the record?

In the example, I need a field the original form filler cannot see. After they submit, I need to be able to go in and fill out this field (invoice number). 

Can I just add a hidden field, then after submission edit this part of the data in JotForm? If so, will this data export along with the original data in the excel file correctly?

I would like to inform you that we have a widget that might be helpful for you. The widget is called Unique ID widget, here's a guide: How-to-Add-Custom-Unique-IDs-to-your-Form-Submissions.

Can I just add a hidden field, then after submission edit this part of the data in JotForm? 

Yes, you can insert a hidden field in the form. In this way, you can add a value in that hidden field when you are in the "edit" mode. Please take note that you need to submit the form for the changes to appear.

If so, will this data export along with the original data in the excel file correctly?

Yes, as long as you export the latest submission. You'll notice that once you edited a form submission, the Grid Report at the bottom will be updated as well.

Here's a screen capture:

Hope that helps. Thank you.