The initial source I’m using for constructing my list of consular posts and personnel is the Register of the Department of State. The Register was printed annually (sometimes more often) and listed the current employees of the Department of State posted in Washington and in the field. Additional contents varied over time, but one can often find information about where foreign governments maintained posts in the United States, brief biographical sketches of DOS personnel, legislation relating to the conduct of US foreign policy, and more.
Here’s the table of contents from the 1900 Register:
And here’s a sample page from the section in which US consuls are listed, also from 1900:
I’m still trying to get a definitive answer as to when the DOS Register began publication, but it was definitely published almost continually from 1869 until 1997, when it shifted to an online publication. As far as I have been able to discern, it wasn’t published in 1881, 1885, 1889, 1890, 1891, 1904, 1905, 1919, 1920, and 1921. (Apparently the Cleveland administration wasn’t too concerned about it? And I suspect the absence in the immediate aftermath of World War I is due to the US government’s failure to normalize its relations with the successor governments of the Central Powers.)
Many issues are available on the Hathi Trust or in Federal Depository Libraries. I have digital copies of the whole run from 1869 to 1942 and will consider requests to share.
For the missing years, I’m using the US Register, which began publication in 1816 after Congress passed a law demanding it. It’s a list of all US government employees. Initially, it was supposed to come out once every two years, but there were some irregularities, and then it became an annual. Many years have been digitized and made available online. (Sadly, at the time of this writing, the Google Books scan of the 1816 and 1818 editions are missing huge sections.) After the Civil War, there were often two volumes in a year, one for postal employees and one for everyone else. The US Register does list DOS personnel, but it focuses on those who drew salaries, so consular agents are not well represented, and all of the other cool stuff that’s in the DOS Register is missing.