When someone chooses to switch into quotes after the word that, you just have no choice but to use quotation marks (despite the fact that your trusty English teacher told you never to use quotes after that). There is, however, no comma after the word that because it really doesn't qualify as "lead-in" words.
...She told me that I had to leave and that "You had better be gone when I get home."
...I said that the report was late and that "Bring it to me no later than tomorrow at 5:00."
Happy punctuating!
Margie