Upvote:2
Since you've narrowed it down to a specific airline, the answer is simple, contact their booking department and ask them directly. They do charge a small fee for booking made via phone, but it is probably worth it if you can secure the flight at the the price you've found.
It is entirely possible that they do now allow you to do this at the price you found. They may view the stops in Dallas and New York as incidental and charge you very differently for the kind of arrangement you are looking for.
Upvote:2
There is an old techinque called the "Open Jaw Ticket" that would get you what you want. Imagine, for example, if you did this:
LON -> DFW (on 1/1) with a return DFW -> LON (on 1/31)
There is nothing that says you couldn't also fly
DALLAS (DFW) -> RIO (GIG) (on, say, 1/3) RIO -> DALLAS (on 1/30).
Throwing NYC into the mix makes it a little more complicated, since your interior ticket now becomes a multi-city, but you get the idea. (I think the exact nature would be LON -> NYC, with an interior of NYC->GIG, and GIG->DFW. Involves one extra leg though...)
Normally, this tactic was used when weekend layovers were required. I used to fly to Detroit every week, so I would fly IAD -> DET on Monday Week 1, with a return of Fri Week 2. For the intervening weekend, I would just have a DET -> IAD round-trip the weekend in the middle.
That said, this technique will guarantee you get the layover you want - just schedule your flights accordingly.
Upvote:3
Almost all travel websites allow you to book a "multi-city" itinerary, where you can enter multiple legs and dates for each leg.
As the trip you've listed appears to be all on American Airlines you could book it on their website (www.aa.co.uk if you're in the UK) by clicking on the "Multi-City" link on the homepage (note that on AA, this is actually a link, not a check-box like the Return and One-way options). You could also use any of the popular travel websites like Orbitz, Expedia, etc.
Most websites allow you to enter up to (at least) 6 flights and their respective dates - in your case you just have 4 (LON-DFW, DFW-GIG, GIG-NYC, NYC-LON).
On the AA site specifically, make sure that you select the "Search by Price" option, otherwise it may end up showing you more expensive flights first.
In practice, many websites Multi-City searches are fairly bad at finding you the cheapest flights - I'm not familiar enough with AA's search to know if it fits into that category, but often it's a good idea to use something like ITA Software's flight search in Multi-city mode to find the exact flights/dates that are cheapest, and then select those exact same flights on the AA website.
If you're not comfortable booking this online then you could also do it over the phone with the airline or with a travel agent, but be aware that they will most likely charge you an additional fee to do this!