What can we do to better level the difference between the poorest americans and the wealthiest?

Because people get compensated in proportion to their ability to produce something of value, in order to reduce the difference between the poorest and the wealthiest, you would have to increase the ability and inclination of the poor to produce more (or I suppose you could retard the ability of the most talented to produce, but what would that accomplish). Compare the standards of living of "the poorest" today to those of the poorest 50 years ago and you will find the poorest FAR better off. Fifty years ago being poor meant you might not have indoor plumbing. Today it means you might have to use a prepaid cellphone.