Can I start NG+ from a previously finished NG+ (i.e. NG++)?

The last time I completed a run of Deus Ex with a regular game (not new game plus), I got the option to overwrite the old new game plus data, however after finishing the game in new game plus mode, I got no such prompt and on starting a new NG+, it started with the same stuff I had when I finished my non-NG+ walkthrough, not what I had when I finished NG+. Did I miss something, or is there really no way to do NG+ with the stuff you had when you finished your previous NG+ walkthrough?

No, it is not possible to start new game+ from a new game+ save. It is only possible to start from a completed normal game.

If you want to change which choices and items you start new game+ with, you will need to make a new (not new game+) save and start the game from the beginning. Upon completing the game you will be prompted whether you want to overwrite your new game+ save.