What should I do to increase the dimming range of LED bulbs?
I'm in same boat.  When I redid the house I'm in, I installed as much LED lighting as I could and the rest is halogen.  What I've learned is, dimming range is based completely on the driver in the LED bulb.  And most LED bulbs will only dim to about 20-30% of there maximum.  

I'm sure you know how dimmers work, but for those that don't it is basically a super fast switch turning the light on and off very quickly.  In incandescent bulbs, this works perfectly providing a nice smooth dimming range.  In LED/CF's it is a nightmare for them to deal with.  To get around it, they actually have to have extra circuitry to interpret the on/off pattern and then use that to control the brightness.  During the on phase they pull extra power to provide power to the light during the off phase as well.  For LED's, the driver reduces the voltage to the LED elements but never actually turns them off.  So what normally would be dimmed to almost nothing for a incandescent, is pretty bright for an LED because it's pulling 2-3x as much wattage during that on phase to keep the capacitors charged and the light going.

TL;DR; For LED's, you have to look for high end bulbs that actually list their dimming range. 

Now hopefully we will not be stuck for long with incandescent analog dimmers driving digital bulbs.  There are a couple of standards trying to gain traction for pure digital dimming.  These work by actually never dimming the power but instead sending a digital signal on the power line to tell the bulb to dim.  Think philips hue bulb being controlled by a light switch instead of a wireless hub.  But right now none are on the market for consumer level products.