First of all, smaller car means smaller cabin but it also mean smaller AC unit. So the question is if the AC unit (and the system in general) is adequate for the task at hand. If you live in a very hot climate, then the best solution might be to look for an upgrade of your AC unit (however this is costly, and you might create more problems to your car if the installation is not done properly).
Regarding your proposal about styrofoam, in a way makes sense in certain conditions. However, the problem is that the air-ducts and stream are -usually- designed to work is a specific manner. Adding blocks in the flow (even though its styrofoam and has little conductivity), it will:
- block the air ways
- in general the styrofoam will act as a thermal battery.
Regarding the last point, I will provide the following comparison:
|
styrofoam |
air |
| density $kg/m^3$ |
40 |
1.225 |
| heat capacity $\frac{J}{kg.K}$ |
1131 |
1012 |
| energy required for 1 $m^3$ to raise 1 $K$ in $\frac{J}{K}$ |
45 |
1.239 |
That means that if you have left also the car in the sun for a few hours, the styrofoam will heat up, and it will act a thermal battery, so it will work against the AC unit. It will store about 30-40 times more than the air, which will be released slowly while the AC is operating.
My advice to you - I live in a place, where last month we had daily max over 38 for 10 consecutive days (some times 40s)-, is:
- when you get into the car,
- open up the windows and let the air cool (the temperature in a car left in the sun can exceed 50 C) for a few minutes as you drive.
- switch on the AC. It will take a minute or so to produce cool air.
- As soon as the AC unit produces cool air, close the windows, and wait for it.
- Air ReCirc: This is probably something very important: You should have air recirculation active, when in those conditions (symbol below)

Other things you can do to help yourself, o get into the car in a hot summer day is:
- park the car in a shaded place
- use windshield sun shades.
- leave the window a fraction open (it will create a chimney effect, that will let the very hot air to escape, and reduce the overheating).
Regarding the styrofoam I would suggest to forego it. Mostly due to the logistics, because even in the once a week that you need to carry something else you will have the problem where do I store the styrofoam). If you really don't use the car for other reasons, carrying other things, consider a bike.