One of these TC-3 Thermo cube units available from AMAZON should help you out.
THERMO CUBEThis comes on at 35 degrees and goes off at 45 degrees. They have different models for different ON/OFF temperature ranges...
Just plug your oil-filled portable heater into it... I recommend the oil-filled heater since this will be used in an unattended area. These units have no visible flame or visible red hot core and would probably be your best bet for an unattended portable heater. Pick the low profile oil-filled heater that is only 15-ionches high if your height space is limited. I believe they also have base-board size oil-filled heaters as well.
Roy Ken