Is it possible to get my existing dimmer switch to dim lights further?
This is often the problem with incandescent dimmers used with LED lights. The lowest setting for an incandescent dimmer is too much when used with an LED bulb.
There are no adjustments you can make to increase the dimming on the actual switch. There are many types of LED bulbs out there so you could experiment with different types until you find the one that fits your needs. You could also replace the dimmer with an LED dimmer. They will lower the brightness on an LED bulb more than the incandescent dimmer will do.