That's a totally valid way of tackling the problem, and there are plenty of other initiatives, such as patent reform or reducing occupation licensing requirements, that would reduce inequality before taxes and transfers. Dean Baker at the Center for Economic Policy Research has a long list of worthwhile proposals along these lines.
But it's worth noting that you can get U.S. inequality down to the levels seen in extremely egalitarian societies like Sweden by doing nothing but changing tax and transfer policies. Pre-tax/transfer inequality in the U.S., as the above chart by the Luxembourg Income Study’s Janet Gornick shows, is about equal to that of Sweden, Norway, and Denmark. Finland, Germany, and Britain actually have higher pre-tax/transfer inequality than the U.S. does. The only reason these countries enjoy such low levels of inequality is that their tax and transfer systems reduce inequality much, much more than the U.S. system does.
And, interestingly, the way they do that is by using relatively regressive but highly efficient consumption taxes to fund very generous welfare states. That's a marked contrast to the U.S. approach of having relatively progressive income taxes but stingier spending programs.