I suppose the sentence you include in your question is correctly punctuated, but it has a lot of other things wrong with it. For one thing, it doesn't seem like a complete sentence but part of a sentence. I have written a sample of how such a clause might be included in a sentence:
He would have been able to buy a car and put some money in the bank, not to mention that he would have been well trained by now.
You will notice that I have substituted "that" for the comma. This seems better to me. Your version might be used in conversation but perhaps not in writing. You don't need to hyphenate "well trained" the way you are using it.
In addition, I don't really like this "Not to mention" phrase, or idiomatic expression, at all. It sounds old-fashioned, like something that Grandma would say.
If you want to use "Not to mention," I suggest you make it part of the sentence that comes before it, either with a comma or a "that" or "the fact that."