When the temperature in a building is already "freezing", does it matter how "freezing" it is?

Solution 1:

From the wiki (edited for brevity):

Temperature Level           Diff  Effect on Health
Comfortable (0°C +)         +2    Cannot become ill
Livable (-1°C to -10°C)     +1    Very low risk of becoming ill
Chilly (-10°C to -20°C)      0    Low risk of becoming ill
Cold (-20°C to -30°C)       -1    Moderate risk of becoming ill
Very Cold (-30°C to -40°C)  -2    High risk of becoming ill (...)
Freezing (-50°C or lower)   -3    Very high risk of becoming ill (...)

The games marks something as Freezing once it reaches a temperature of -50°C or lower. Anything below this, is still Freezing, and subject to the same effects. Essentially, the game works with Heat Level Differentials, ranging from +2 to -3, where -3 is the worst it can get (Freezing).

So to answer your question: no, it does not make a difference if you make your buildings not that far below freezing point. You will want to reach the previous level of Heat Differential (-2, or warmer than -40°C) to get any change in effects.