I also use an oil-filled radiator type heater for supplemental room heat.
I have a setup I really like. I found a thermostat rated for 1500 watts at a farm supply store. I believe they called it a thermostat for "milk house heaters". It was only about $30 several years ago. But really any high wattage thermostat would work. Mine is just at the cheap end of the spectrum.
I plug a multi-strip outlet into the thermostat and then plug the heater and a large box fan into the multi-strip. Obviously what this does is cause both the heater and the fan to kick on when the thermostat cycles on, therby distributing the heat around the room. I could leave the fan running all day but prefer to use it as an audible indicator of how often my thermostat is cycling.
To be even safer I also adjust the wattage output controller on the heater. My idea being that if the thermostat fails I won't have the heater set for too high of a temp and endanger my animals. This can take some time but worth the peace of mind for me.