Birth Abroad Out-of-Wedlock to a U.S. Citizen Mother:
A person born abroad out-of-wedlock to a U.S. citizen mother may acquire U.S. citizenship under Section 309(c) of the INA if the mother was a U.S. citizen at the time of the person’s birth and if the mother was physically present in the United States or one of its outlying possessions for a continuous period of one year prior to the person’s birth. The mother must be genetically related to the person in order to transmit U.S. citizenship.
If I’m not mistaken, that clause reads, or at least read back then, that the unmarried U.S. citizen mother could transmit her “nationality” to her foreign-born baby.
“Nationality” and “citizenship” are two distinct legal terms: while all citizens of the U.S. are also nationals of the U.S., not all ationals of the U.S. are also citizens of the U.S. For example, American Samoans are nationals of the U.S., but not citizens of the U.S. Nationals of the U.S. may travel to and (I believe) work within the U.S., but do not have rights reserved for citizens (such as voting in U.S. elections and passing on U.S. citizenship via jus sanguinis).
Thus, unless I’m missing something, an unwed mother would pass on her U.S. nationality to her foreign-born son who was not a U.S. citizen at birth under some other law, which would allow her child to live and (eventually) work in the U.S. But she wouldn’t pass on her U.S. citizenship.