First of all, since it sounds like this could be considered "occupied space", you better check your local building codes and/or with your local inspector to see what is allowed. Most codes have specific requirements for stairs, and in some jurisdictions strange things like spiral stair cases may not be allowed or may require specific approval because of their narrow treads.
I'd definitely look to see if there is any way to get more space at all. For example, what is behind the closet? If you can lose a bit of ceiling height there, you could get a bit more room for the staircase. Likewise, is it acceptable to have the stairs coming out into the room at all? This too, will provide a bit more space. Using this calculator, if you can do 10 rises of 8" (this may be steep though), you need 6'8" of total run space.
![enter image description here](https://i.stack.imgur.com/b1NkW.png)
Depending on the adjoining space, you might also be able to make use of the room under the stairs.
If you can't get any extra space, you might consider a "ladder staircase" (kind of a hybrid of ladders and stairs):
![enter image description here](https://i.stack.imgur.com/PrFV7.jpg)
There are some interesting designs for these, including alternating treads:
![enter image description here](https://i.stack.imgur.com/u8jGO.jpg)
And some really funky variations if you want to be really creative:
![enter image description here](https://i.stack.imgur.com/Fn1AO.jpg)
It's a balancing problem, fairly typical for a one-zone heating system spanning an entire house.
Mind you, I'm a bit concerned with a setup that kicks on the A/C in heating season - being from a primarily heating climate, we typically have a "winter/summer" or "heating/off/cooling" mode switch either on the system/boiler/furnace or on the thermostat to prevent that sort of foolishness. If it's REALLY so hot in winter or cold in in summer that we want to change the mode, WE choose to change the mode, rather than leaving it up to a machine to decide.
Presumably you have forced air heat, since the same system is cooling. One approach to improve balance is to run the fan more to distribute air around the house and balance temperatures even when heat is not being delivered. A more basic step is adjusting the airflow to different parts of the system for heat delivery that more closely keeps things even - but if the same system is cooling this may be difficult to get balanced in a manner that works well for both, since heat rises and cold air sinks, left to itself.
The system may be oversized (so it quickly heats or cools the location of the thermostat, and then shuts off the fan, rather than running a large percentage of the time when it's cold out), but all systems are prone to being somewhat oversized much of the year in order to be large enough to heat on the coldest days and cool on the warmest days.
A good HVAC professional may well be worth talking to in order to tune your system as best it can be tuned. Moving to continuous circulation (perhaps at a lower fan speed) seems like it might be needed in this house to reduce stratification, at a guess. The only major downside if your system is not too noisy will be the electric bill for running the fan.
As a quick stopgap, examine your thermostat to see if it has a "fan" switch, typically with two positions - Auto (blows only to heat/cool) and "on" or "Continuous" - if the switch exists and is wired correctly, that should put the fan in continuous circulation mode - but you may want to alter the system to make that quieter and/or more efficient with a lower speed or even an entirely different fan/blower. And it may still need to be balanced to work better.
Best Answer
In summer the attic is heated by the sun through the roof and windows and also by warm air rising from the rest of the house through the staircase. Closing the attic door would reduce the amount of warm air rising, but that is not the main source of heat. The sun is. And closing the door also decreases the flow of air from the attic registers back to the return on the second floor. The gap below the attic door is just not enough to conduct an entire floor's airflow back to the return.
These were the problems:
These were the steps I took:
So far so good. Within 5 minutes of the system starting up, all the rooms on both upper floors were in balance and when the thermostat became satisfied all the rooms were at the set point. This worked with the attic door open or closed. Once the system shut down the attic rooms would then heat up much faster than the second floor so by the time the thermostat called once more for cooling, the upstairs was about 10 degrees too warm. So .....
Notes
This was not a DIY project for me. The ductwork changes took a pair of professional installers an entire day. Because I had a finished attic the job to move the thermostat from one interior 2nd floor wall to another one was a bear of a project for an experienced electrician with a helper.
It was easy for me to install the attic return because the upstairs air handler is in the unfinished attic space behind the wall of the hallway. They just had to poke a hole in the wall and add a short duct that joined the main return in the plenum outside the handler. If the handler were in the basement it would have been a MUCH bigger project.