0.001 USD is called a mill.  It did find some use early on in the history of America, and it still can be found in American gasoline refilling stations (where the gasoline price shows an additional 9 mills per gallon).  It also applies to taxes where sometimes taxation rates are set in terms of a number of mills per thousand dollars of value.

Considering that the penny is now threatened with extinction due to inflation, it is obviously a seldom used denomination.  If the dollar ever went deflationary again over a prolonged period of time it might be used again, but that wouldn't happen as long as the Federal Reserve Bank is in charge of monetary policy in relation to the U.S. Dollar.