Had the opposite problem on my 83 300D turbo. The light came on any time it wanted. I finally replaced the in tank sender, I tried to clean it but it did not last. In case you don't know the tank sensor is what triggers the light to come on bright indicating low fuel or reserve. Parts shop sells the unit for about seventy bucks shipping and all. it was worth it in piece of mind, My wife had run out of fuel once and that was enough. you can get to it through the first aid kit behind the rear seat, just take the two screw's out and lift the kit out. Then take a pair of channel locks and unscrew the unit. It should not be to tight, you will also need to reuse the o-ring from the old unit. do not overtight. I do not know of anything else that would cause the light to not light up at low fuel. As you said earlier the bulb is ok.
2006 E350 98K miles
2013 Ford Explorer 15K miles