arduino - Expanding Range Of Analog Values From Temperature Sensor

I'm trying to build a temperature controlled relay module with an ATtiny45 ( Arduino UNO R3 as ISP ) and a NXP KTY81-222 Temperature Sensor.

The temperature range i want to monitor is 20°C to 30°C.

( data sheet -> http://www.produktinfo.conrad.com/datenblaetter/150000-174999/153653-da-01-en-TEMPERATUR_SENSOR_KTY10_7_KTY81_222.pdf )

My circuit layout looks like this = ( I'm using the Arduino UNO R3 for debugging )

**notice that i've also tryed to "swap" the 1K resistor and the Sensor position to invert the output.

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*and the arduino board ->

A0 is the Analog Input 0 of the Arduino. 5V is directly plugged into the 5V Output pin of the arduino.

( later it would be the same on the ATtiny45 using some batteries )

( VCC can be 2.7V to 5.5V )

I've used a household digital thermometer to compare the values from the analog input with the measured °C values.

this is a quick measure table =

Analog -> °C

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687 to 690 -> 24.4°C to 25.1°C

707 to 708 -> 36.2°C to 36.5°C

as you can see i have a range from about 20 Analog units that represents a range from about 10°C units.

so what i actually want is to "grow" the range of the analog units to about 200 Analog units for 10°C units range. the solution should be a good to handle value range, so i can tell the µC to power the relay if temperature is under 24°C, and close the relay if the temperature is over 28°C.

I want 20°C to be an analog value from about 500 and 30°C to be about 700. ( this "range from 500 to 700 would be good enough for me.

i hope i could explain my question well for my bad englisch.