You don't get to put out any more cards with the 2 Time you just spent to Explore. You can choose an existing door to Explore, still, and can choose to Flee or Encounter the card.
Once the Explore action is complete, you can decide to go onto the next level, or spend more Time (adding cubes to the Stairs) to Explore the existing doors again.
Once you go to the next floor/level, then you gather all the left-over doors and the discard pile to shuffle into a new draw deck.
In our situation we had no rooms (open or closed) left and only 2 cards left on top of the Stairs. So those are pretty much pointless? I see there is no point Exploring the current floor. We figured we could use them onto the next floor to explore but it sounds like we can't.