I have an earlier answer that might interest you!
Joel Dinerstein tells the philosophical and cultural part of the story in The Origins of Cool in Postwar America. Although he doesn't use these terms exactly, he argues that "cool" entrenched itself in English vocabulary because its surface meaning in the 1940s tapped into a deeper, underlying ethos in American and European society that transcended the specific, narrow manifestation of its original useage--and the specific, narrow set of people to whom that manifestation applied.
The English language (and many others), of course, has a rich tradition of using temperature metaphors for human behavior. "Hothead" for a quick-to-anger person goes back to the 17th century; "cold" and "cool" to describe someone emotionless or dispassionate--think "ice queen" or a "glacial" expression--are already found in their Old English equivalents! The concept of coolness, however, is more recent.
This use of "cool" emerged into wider consciousness in and through jazz musicians and music in the late 1930s and 1940s. In particular, black jazz artists deployed the term to define a certain sensibility, a way of playing music and of being in the world. "That's cool" or "I'm cool" or "keeping cool" united the ideas of control, relaxation, intensity, individuality, and conscious presentation/performance--against hostile external forces. (This follow-up question explains why contemporary African-American jazz musicians found particular power in hot/cold metaphors.) In the songs of black musicians of the 1940s, like "Keep Cool, Fool" (Erskine Hawkins) and "Stay Cool" (Count Basie), the significance of "controlled, relaxed intensity" in the face of toxic white culture stands out starkly: sharper-than-glass public composure and conscious, rational self-creation fly in the face of the range of pop cultural stereotypes of African-American men--the Angry Black Man, the peppy and dim-witted minstrel figure, the obsequious and self-negating Uncle Tom.
Thus, the manifestation of "cool" by black jazz musicians had a contextualized social power specific to black culture, and especially black masculinity. But Dinerstein argues that as the word cool leeched into broader (not just black, not just music-related) useage, people glommed onto not its surface meaning but its underlying one.
Although "cool" first emerged in the late 1930s, Dinerstein argues, the instrinsic message was exactly what the traumatized post-World War II, late imperialist West wanted and needed to hear. "Cool", on a deeper level, was the conscious construction of self as rebellion--a "public mode of covert resistance."
He ties the importance of this to a number of cultural currents in 1940s-50s America (and to a lesser extent Europe--America is his primary focus). World War II was an existential crisis/turning point for the West, an undeniable marker of the failure of its collective ideologies to necessarily make the world better instead of worse. In addition to realization of the horrors of the war and the Holocaust, the time period marked the earliest consciousness of Western white power that maybe Western imperialism was an inherently bad thing according to colonized people. And specific to the NATO bloc, you might call it, individualism was a politically promoted and powerful tool against the anxieties of nuclear war with Communists.
There was also further proddings at questioning a rock-hard system of necessary Christian virtue versus accepting the darker side of human emotionality to a limited, productive degree. And of course the war and its aftermath brought about a massive crisis in masculinity. (Masculinity is always in crisis, and masculinity in crisis is always causing things.)
Dinerstein traces early specific manifestations of this underlying sentiment through multiple Venn diagram circles of Western culture: black male jazz musicians, white male film noir protagonists, and white male and female existentialist philosophers (no, really). This is a fascinating, and more importantly compelling, choice of cultural nexuses because they appear, on the surface, so distinct from each other. But that is precisely his point: a given appearance of cool is almost beside the point. The underlying sense of "conscious construction of self as rebellion, without people realizing it's an act" was the meaning that people took--and more to the point, take--out of "cool."
There are some really interesting asides throughout the book that help convince me Dinerstein is on the right track with his philosophically-tinged cultural analysis. One point I really liked was the emergence of "chill" and "to chill" out of black culture in the 1980s. In other words: a time when and a group of people for whom to choose to relax, to choose to opt out of Production had become an act of social rebellion against welfare queen/entitlement rhetoric and the escalating societal value on giving everything to work.
I do think there's a second layer to the story here: the technological (in terms of engineering technological, but also social technology) reasons that a word emerging in a music subculture at the end of the 1930s would have New Yorker essays dropping it casually by 1955--essentially, the reasons that Parisian philosophers and Southern black musicians and Rust Belt white factory workers could connect to an underlying meaning in a word because their cultural orbits were common enough to be mutually intelligible. I would like to see some analysis of the role of radio and early television, especially, in streamlining transmission of a nebulous idea across what we today would see as disparate social groupings.
But overall, his portrayal of the underlying meaning of cool through its surface manifestations resonates even with our current, memetastic version: Look, look at all the fucks I give.