Do any home thermostats have an adjustable control range?
"Swing" is the term you want to look for. Some thermostats allow you to adjust the swing which usually changes when it comes on and turns off.  For example, say the swing is set to 2 degrees and the thermostat is set to 72.  It will then turn on at 73 and cool to 70 (two degrees cooler than set temp) then stop.  Some thermostats might use a number that doesn't directly correlate to degrees, but it works similarly.

If you look at the manual for this thermostat it says:


  TEMPERATURE SWING
  A thermostat works by turning your heating or cooling system on and off whenever the
  room temperature varies a certain number of degrees from the set-point temperature. This
  variation is the “swing.”
  Your system should cycle on about 3 to 6 times per hour. A smaller swing number
  increases the number of cycles, so room temperature is more constant. A larger swing
  number decreases the number of cycles, to save energy in most cases.


It seems that it uses a number from 1 to 9 that doesn't map to degrees.

I just looked at the help page for the smart thermostat I use, and they don't use the word "swing", but instead have a variety of "thresholds" that are adjustable.  So I guess that is another term you can look for.  I highly recommend the Ecobee thermostats, but they are pricy and don't make sense unless you really want high tech.