Open-Jaw flights (tickets) are great option for independent travellers:
An Open-Jaw (OJ) airline ticket allows you to fly out to one destination but return home from another, giving you the freedom to plan a route without having to worry about returning to your point of arrival.

On my last big trip to South America I had an OJ ticket; I flew out to Ecuador (Quito)
then travelled overland to Brazil, via the southern tip of Argentina. A great trip, which took just over 4 months. My return flight to London was from Rio De Janeiro.

OJ tickets can also work well for shorter trips; e.g. a three week trip, flying out to Lima Peru and returning home from La Paz, Bolivia would be a good, whistle-stop introduction to South America.

Because of the vagaries of overland travel, you should buy your OJ ticket with a flexible return date, flexible in this context, means giving a nominal return date when you purchase the ticket, but having the ability to change that date, for free, with a phone call to the airline when you’re out on the road. This flexibility has limits however, an OJ ticket will have a maximum stay, e.g. 6 months, requiring you to return home within that period. The longer the maximum stay on the ticket, the more expensive the ticket will be, so choose carefully.

An Open-Jaw ticket with a long maximum stay gives you the freedom to take a lengthy diversion or extend a volunteer stint en route, the flexibility works both ways though, allowing you to bale-out of your trip early if required; a friend of mine injured his knee sand-dune surfing near Pisco in Peru, although not seriously injured, he opted to bring forward his departure date, take a local internal flight direct to his departure point (Buenos Aires) and jump on a plane home.

For those people travelling from Europe, I’d recommend flying to South America (SA) on a European airline such as KLM or Iberia, as opposed to flying on US-based airline. Why? because when flying with a European airline your stop-over will be in Europe,
e.g. KLM flights stop in Amsterdam, Iberia stops in Madrid. So with European airlines there is a short flight followed by a long flight when travelling to SA: In my case, London to Amsterdam, duration 1h 30m, then Amsterdam to (say) Lima, duration 15 hours.
I find it much easier on the brain and the body to have a short and a long flight, rather than two medium length flights of ~6-7 hour duration, which is what happens when you fly with an American airline via the USA.  I find the stop-over between mid-length flights – when you literally don’t know what day it is – to be purgatory 😉