@ spookyB you are correct. Don't complicate it by sending another letter. I am sure this is a common scenario, especially with people who work in big companies where head office is located separately. (head office usually issues T4s). You will be fine
it's the same in my case, but the official letterhead for my company has all the branch addresses, including headoffice, listed in the footer of the letterhead paper). For me, it gets complicated because my company was bought out on the day my boss wrote the letter (he did include this information about the impending takeover in the letter and attached a press release confirming the same, and indicated that the transition from company A to B is still in process)
So I am just as anxious as you. I don't know how it will play out.
Lets just hope we prevail, keep me posted on your case!