One can only imagine what it must have been like in the smoke filled rooms in the back offices of P&W back in the 50's when all this was going on.
On one side of the room you have the cigar chomping military sales guys stating what the latest paper airplanes were being drawn up in the far off skunk works, and that millions of development money would be there for the asking if P&W would offer up product designs on paper to match the paper airplanes the government had already decided to fund for study.
It wouldn't have taken much for the P&W advanced engines groups to dust off a previously desgned and protyped model, add some stages, and afterburner and then claim that they could easily extropolate a performance and thrust size increase based on old data and with a few million actually commit some componet (scale hardware) on rigs to verify the feasibility in a step funded program.
Then on the other side of the room the fleldgling commerical applications dream merchants were saying that some of the military big guys like Douglas and Boeing were thinking of entering the jet age if an engine might be avialble in 5 years or so.
Not to worry it would be relatively easy to move some accessories around take off an afterburner and offer up paper model like the JT3 series and then a b then c as the specs tried to keep pace with the same dreams in the commerial sides of Douglas, Lockheed and Boeing. Actually with enough development money it actually worked as the Collier trophy for the J57 attests.
By the time the commecrial airplane guys finally settled on a plane that matched the available thrust to weight, then the various development models finally settled on a compressor and turbine combination that would match and a salable model would actually be sold in quantity like the JT3c
This went back and forth through the 50's and other models jumped in like the J60/JT12 and the J52/JT8 J58/JT11, even though the actual sales numbers might actually have been low and the model itself stored as a development only machine for future possibilities.
So when the first concept of a commercialy viable fan came along all they did was trot out a bigger compressor blade and bypass part of the air and calim better fuel consumption and quieter noise4 signature. While at the same time changing the4 model number by one charcted from a "C" to a "D". This was such a hit with the JT3c moving to a JT3D that they figured they could do the same thing with the orphaned JT8 by adding a new compressor and a fan and since the "D" change meant oodles more performace they just decided that was te way to desgnitae any ne4w commecial engine with a fan and two separate compressor sections.
Now as for the mysterious J57, J60, J58 designation you need to ask the airforce how they came about in their smoke filled back rooms. Remember since the government was spending their money their were a lot of number combinations that existed on paper both in the government offices as well as the engine manufacturers design and component development groups..
Some of these combinations even had hardware like tiny turbines behind boron fuel tubes and gears between the turbine and the large compressors, with afterburners 6 ft in diameter. I never heard of one finally coming together as a full engine though.
One of my neighbors worked there in the test department straight out of college, but sadly he died just before this thread was started so I can't trot over there for a beer and questions anymore