From The Lambert Conformal Conic Projection With Two Standard Parallels:
The standard parallels are usually chosen at one-sixth and five-sixths of the total length of that portion of the central meridian to be represented.
Are there any similar rules for Lambert Conformal Conic with one Standard Parallel for choosing Latitude of Origin/Standard Parallel for a given region?
The latitude of origin parameter is not a standard parallel. In a two-standard-parallel case of Lambert conformal conic, it is used to identify the origin of the northing / Y-coordinates only.
In the scale factor version of Lambert conformal conic, you specify the latitude of origin (or sometimes latitude of center) plus the scale factor. Mathematically, the scale factor, if less than 1, means that there are two implicit standard parallels, not quite symmetric around the latitude of origin.
For this second case, I have not seen a useful approximation for setting the scale factor parameter. I think you would need to find a program that reports the point linear scale factor (like the Tissot Indicatrix values). Set up the custom Lambert conic coordinate reference system, and then project some points to and see what is reported.
Or you could write your own program, that given the defining parameters, what are the implicit standard parallels?
I wrote an program to do so using John P. Snyder's Map Projections:A Working Manual several years ago. You'll want to look at book pages 107-8. On 108, he mentions how one equation is indeterminate if standard parallel 1 == standard parallel 2. My program is quite stupid, it just calculates a bunch of 'k' (scale factor at point) values given inputs ranges and I repeat the program, narrowing down the range until I get k=1 for one of the implicit standard parallels.
