We've already discussed the basics of adding a free one-way flight before or after a round-trip award reservation on Delta and on United. Free one-ways on American Airlines are a little trickier than on either of the other traditional carriers. As always, in order to book a free one-way flight, you need to use a stopover. However, American only allows stopovers on award tickets at the "North American gateway city," which is the airport where you depart or arrive North America. This post has a list of which cities are considered North American gateway cities with different American partners.
Therefore, the only way to add a free one-way to the beginning or end of your itinerary is if you live in your North American gateway city, or can get there cheaply or easily. For example, if you live in Boston, then as long as you depart or arrive North America on a flight from or to Boston, you can add an earlier or later one-way flight at the beginning or end of your reservation. Below is a simple example. On the outbound international leg, Boston to London, Boston is the North American gateway city. That means that I was able to add an unrelated, free one-way flight from Dallas to Boston at the beginning of the itinerary for the same miles as just the international round-trip:
On the return flight I found, Chicago is the North American gateway city, so it would be possible to stopover there if I hadn't used the stopover already at the beginning of the itinerary. Of course in order to get from Boston to Dallas in the first place you'll need a paid one-way ticket or an award from an airline that allows one-way awards at half the price of a round trip, like United or American.