I don't know if I'm allowed to respond here because i'm not an official expert or whatever, but... In the specific case of Radar putting through a call back home to the states, the field telephone at the front would likely be connected by wire to a switchboard behind the lines which aggregated wired lines from a number of forward posts. At the switchboard an operator would patch it through to an HF radio facility capable of maintaining a contact via HF (what we'd now call shortwave) radio to an intermediate station or even directly to a stateside station, where another operator would establish a telephone call on the US telephone network, and "patch" the HF radio connection through to its destination.
Because HF radio waves propagate through the atmosphere differently based on time of day and even to some extent year, the HF radio facility would require a variety of large antennas and its operators would require some amount of skill and luck to be able to maintain a reliable connection for any length of time. It was a substantial task.
The use of HF radio was more or less entirely replaced by satellites once they became viable since they're predictable and reliable and don't require giant antennas or any amount of luck. In the late 1950s submarine telephone cables came into use and became the predominant means of peacetime long-distance voice communication.
Switchboard of approximately appropriate era
Some discussion of USSTRATCOM facility in Davis CA, one of the relay facilities. Many of the relay facilities handled more data (teletype) communications than voice, but over long distances were carried over the same signalling systems (ie, transmitting binary signals using audible tones, one tone for zeroes, another for ones) known as voice-frequency telegraphy. http://californiamilitaryhistory.org/CommStaDavis.html
Had to do some digging to match up with what I remember from school, but in the early 50's a lot of it was VHF, microwave, and long haul cable.
http://www.armysignalocs.com/veteranssalultes/sig_corps_korean_war.html
The biggest benefit was a pretty extensive cable system called the Tokyo-Mukden Cable:
http://www.armysignalocs.com/veteranssalultes/SCS_Mukden_2.html
"Originally built by the Japanese through slave labor from both Korea and China, on its opening on September 30, 1939, it was the longest communication cable in the world, stretching some 2,700 kilometers along its full length."
Now, as for how the call got from Japan to the US, I have no idea. Maybe routed through Europe?
If there were no direct phone lines, then a local radio phone signal was picked up by a repeater tower in close proximity. The signal was re- brodcast by the repeater to a communication station where the signal was connected into a telephone line wich could be connected to the U.S
