Can I use dimmer voltage regulator for single phase multi speed AC motor?
A 2-wire overhead fan controller would work. You can keep the 3-speed switch and put the speed controller in series with the lowest speed tap. That way you have all the higher speeds, and the LOW becomes a LOW-LOW.
In your photo, the heat sink seems small. If it gets too warm, add some metal or put the dimmer into the airstream. Remember, don't go too low on the speed setting. But if your motor is already turning when you switch it to LOW-LOW, then the setting can go a bit lower. It's like trying to drive your car using second gear. It helps if the car is already rolling.