There is no basis for wearing one head covering that has two layers. There are some sources that suggest that, while davening, it is not considered proper to simply wear one’s hat and discard his kippa – a person should also wear his kippa underneath. (The sixth lubavitcher rebber, in his igros, suggested attributing kabbalistic significance to this).
That is how the “two layered kippa” story started. It is a bubbamaysa that was concocted during the 20th century.
In general, the earliest source for specifically wearing a hat during davening (as opposed to a smaller head covering), is derived from the Rokeach. It is proper to have atifa during davening, such as by draping a tallis over one’s head. In 13th century Germany, people did not wear a tallis during mincha or ma’ariv. The Rokeach suggested that the people were at least accomplishing “miktzas atifa” (partial atifa) by wearing their hats during davening. This Rokeach was quoted by some later achronim, but was not specifically mentioned by a large number of poskim.
As the Mishnah B’rurah writes, wearing a hat during davening is primarily a means of “hikon” (proper decorum during davening). After all, as the Chofetz Chaim writes, if people wouldn’t even walk down the street without their hats, it is certainly disrespectful to daven without a hat.
The basic requirement of hikon is that someone present themselves at davening the same as if they were going to a meeting with a governor or president.
In communities where people walk down the street without a hat, and go to job interviews without a hat, and even meet the president without a hat, “hikon” doesn’t require a person to daven with a hat. In such a case, the only possible halachic basis for an ordinary Ben Torah to wear a hat would be the Rokeach. (Of course, if someone is davening in a shul where everyone wear hats, “lo sisgod’du” and “minhag hamakom” may also be factors.)