"not to mention that" is a correct and usable phrase in written English. This phrase is typically used when adding another point to an already existing list of items or ideas. For example, "She had anger issues, not to mention that she also had severe depression.".
Not to mention that hockey is hard, really hard.
(Not to mention that Karzai seems more Fredo than Michael).
Not to mention that Rick Lazio is a New Yorker.
"Not to mention that right now it's a rental property".
"Probably best not to mention that".
Not to mention that it's a lot of fun.
(Not to mention that cash).
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union