Another possibility worth checking on, some HVAC units have separate piping going to different levels. In these the best approach is to put the upstairs wide open and the downstairs mostly closed in the summer and then reverse in the winter. That will help fight your problem more and balance the house out a fair bit even with just your current thermostat, worth checking to see if you have this option. In the house I saw this in it was the piping coming straight out of the HVAC unit that had this and you could close and open with little metal bars on the sides that rotated.
Beyond that, are you more concerned with evening out the house temperature, keeping your costs low, or both? Your post doesn't say which is the priority and it would make a difference.
Finally, it's also well worth looking into some simple things you can do to reduce the impact of the sun. Old southern houses used to be built with things like that in mind and it helped keep houses cool long before air conditioning. Closing shutters, curtains and blinds to keep the sun out is a big one (you'd be amazed at how much difference this can make), if the outside air is cool enough in the morning and evening then opening up doors and having a big fan pull in the cool air at those two times can help, as could just a fan on the stairs pushing air up to mix more. Are you concerned the upstairs windows are letting in heat or is it just the light heating? If the former maybe better insulated windows would help? I'm sure a few google searches along those lines will tell you a lot.
Might be tempting to go out and buy a new thermostat first things to fix this but before you put money toward a new product I think it worth checking to see if a little ingenuity, elbow grease, and internet knowledge could fix your problem with no cost to you! A fun challenge to try to fix.
Followup Thoughts: Also, it struck me, even if you don't have the option to shift your airflow to the upstairs, you could just block off a few of the downstairs vents manually (cover them with tape or something) to increase the pressure in the system to upstairs. Another potentially easy fix that won't cost you any money.