I have been Flying for over 25 years and I have never heard of such flexible program that allows a person to take a break in between long flights.
One of the biggest problems with long distance flights with multiple stops is the dreaded layover in different airports. And just sitting around waiting for the next plane.
I was sitting next to a couple who were flying from LAX to Rio through Panama City. I was on flight 473 from LAX and the big talk around about six passengers was the 3 hour layover in Panama City and then back on another Copa flight to Rio. Most couples in this type of situation are vacationing and so using the "Stop Over" program the couple could have stayed a couple of night in Panama City and proceeded to their next destination. I am sure if they knew about this program, they would have been delighted. Here is the more about the program below:
Through the alliance between Copa Airlines and the Panama Tourism Authority, is being promoted STOP OVER program, aimed at travelers of the United States and Latin America whose final destination is not Panama by allowing said passengers to stay in Panama for several days without receiving any penalties for changing their flight itinerary.
The travelers just need notify the moment the ticket reservation is made.