I take Dianette, which is the UK version of Diane, and it's only when you've missed more than one pill that you have to start a new pack. A pill has to be more than 12 hours late before it can be considered 'missed', so you have some leeway if you accidentally forget. (just checked info for Diane35 and it's the same) The only time it's recommended to start a new pack is if you miss an active pill in the last seven days of the active (usually beige) pills in the pack. If that happens then you carry on with the pack, but instead of taking the placebo pills you start a new pack of active pills instead and skip your placebo week.

In addition, you can start it now. The progesterone your doctor prescribed won't give you a period anyway, just a withdrawal bleed. If you start the Diane now it'll be seven days until you can safely have sex without using another method too. I use Dianette for PCOS (for about 14 years now) and took it without having a period on three or four occasions.