Insulate behind fixed walk up attic stairs

Question

I have a full set of stairs that walk up into an unfinished attic space. The full sized access door to these stairs is in a hallway as if its a closet. Unfortunately, the max distance I have under each stair before I hit the attic-side of an interior wall (solid blue line in picture below) is only 6 inches. What is the best way to insulate this under the stairs so heat isn't lost from the interior closet pictured? I've drawn a diagram below:

One idea I have is to add rigid foam on the inside of the closet as FreeMan has recommended. But what can I add to the big void under the first 3 stairs and what about the small area behind the higher stairs?

Here is a close up shot of under the bottom 3 stairs. The prior owner put light insulation in place with the face on the inside of the attic and barely any insulation to fill the giant void:

I'm wondering if I put faced insulation against the drywall of the closet. If so, how do I secure it? There are no lumber beams to staple to. Then in front of that I loosely stack a few batts of unfaced insulation to fill up the big void?

If you are curious about the sides of the stairwell going up to the attic: they are insulated with fiberglass insulation in the wall beams and then a 2-3" rigid foam insulation in front. So when I walk up the stairs the sides are shiny silver like I'm walking into a spaceship.

Answer 1

I'd consider adding 2 inches of foam board to the closet ceiling, then drywall. Finish as desired.

Answer 2

Best way is to add a door at the top to stop airflow and add insulation to that door. On the other side - insulate the door leading up to the attic. There are cheap door insulation kits sold for doors leading to garages... same thing here.

If you don't go the route of stopping drafts I would insulate door and add padding and carpet to stairs. Stairs notoriously let in air. Some cheap carpet would not only insulate but be an air stop.