With the modern Airlines fuel efficiency wich almost brings the plane around the Globe on the initial fuellling, one has to think hard or use the imagination to figure all the refuelling-stopovers wich long-distance flights demanded back in the 50's.
SAS was the first to use the so called polar route for the Scandinavia-to-US Westcoast destinations. However, in '56 it was with the propeller driven DC6 wich demanded some refuelling stop-overs on the trip: Sdr. Stromfjord, Greenland and Winiepeg, Canada for the westcoast and for the eastcoast it was Keflavik, Iceland and Gander, New Foundland - CAN.
As late as 25 years ago did my vife fly Copenhagen-Miami with a DC10 wich refuelled in Gander, however I think that Gander at that time mostly kept itself alive due to low fuel-prices!
When I writes all this it's because the Europe - Far East trip must have had a likely fixed route with established refuelling stopovers, wich probably didn't change much until the modern long-distance Airlines was able to fly non-stop.
Many of these refuelling-airports was probably only primitive airstrips in the middle of the desert, and Your memories got the tell-tale signs of such due to primitive sheds, minimal lights and fuel in barrels!
There must be someone who knows this regular route and by the help of Google can find the fuellling stop-overs!?
It would be much helpfull If You could tell from where You travelled and wich destination You had!?
Anyhow, I think that among European airlines had the british companies the lead when it came to far east connections and You should surf for british airline logo and colours of the 50's to see If You can be able to reckognize the colouring of the plane to determine wich airline, and thereby wich route the talk is about!
Without any knowledge of far East routes, I can imagine that Iran (Persia, at the time) could have been convenient for refuelling too :-/
Finally I have to thank You for the description of the long-distance travellers troubles at that time: Stop-overs in the middle of nowhere without any accomodations outside of the plane is totally unimaginable for a modern globetrotter :-o