Unstable analog sensor readout
the sensor gives unstable readouts varying for around 0.03-0.04 V each
  readout cycle


If your sensor outputs 0 to 5V and you're only getting 0.03 to 0.04 variance then I would say that it's working pretty well.  You can get a more stable output by averaging several readings before printing them.  

Additionally, for even more stable readings consider powering your Audrino from a battery.  USB power can induce a considerable amount of noise if it's coming from a PC or Laptop USB port.  Also, there is a low noise mode where you can put the Arduino to sleep during your ADC reading (and it wakes up afterwards) which helps a lot with ADC noise reduction.