Page 19 of this:

http://klabs.org/history/history_docs/mit_docs/1711.pdf

from 1976 is likely as good as it gets without really diving deep.

It doesn't have hard dates on when software development started (but likely coincided with the early engine tests in the early 1960s)

It begins:

    The very early programs for the first few unmanned earth orbital test flights were each put together by a small dedicated group led by a chief engineer-programmer. 

It then discusses many software components of getting a rocket from the earth to the moon and back again and the various people that did the over arching thrust of each software component.

It then moves on to say:

    Much of the detailed code of these programs was written by a team of specialists led by Margaret Hamilton.

    The task assignments to these individuals included, in addition to writing the code, the testing to certify that the program element met requirements.

    Overall testing of the assembled collection of program elements necessarily took the use of considerable human and machine resources. 

So it's a big team effort (no surprise) taking many years, with key players for each component and a seperate group tying all the pieces together and bringing them to standard, to mesh well, and ironing out the hard stuff (fine details).

Hamilton's SAGE work ended 1963, Smithsonian asserts she started and later was promoted in 1965.

Your claim is that the Smithsoniam got this wrong?

By her own words, yes. I think the Smithsonian and Wikipedia are stuck in a game of telephone. Working on "the least important software" during the unmanned Apollo missions, but also already leading her own team by 1965 are clearly incompatible.

She also references lunar landmark tables, which wouldn't have been relevant until missions that started in '67-68, so she's not talking about earlier unmanned suborbital tests e.g. AS-201.

https://en.wikipedia.org/wiki/AS-201

I can't find any primary source suggesting she worked there before then - her name doesn't start appearing in document dumps until January 1968, e.g.: https://www.ibiblio.org/apollo/links2.html

Maybe that's indicative of when she got promoted, but given that she painted herself as a complete newbie around the same time period, that's quite the career trajectory.