The simplest rule-of-thumb, one that will rarely lead you wrong, is always to avoid comma after "So" at the start of a sentence, immediately after a semicolon, or immediately after a comma. That applies regardless of what "so" means there or what grammatical term you think would describe it.
The more nuanced answer follows:
Polished writers who grew up before approximately the late 1990s, reading literature written in the 1800s and 1900s, rarely place a comma after "So" in the sense of therefore. It looks and sounds as out of place to us as a comma after "But" — which has also become all too common in the last 20 years.
The de facto authority for the vast majority of Freshman Comp classes, The Hodges Harbrace Handbook, added an explicit rule for this reason, Rule 13b: Avoid comma after coordinating conjunctions: For, And, Nor, But, Or, Yet, So. (I've tweaked the order to correspond to the now more common acronym FANBOYS. Google FANBOYS comma for more perspectives.) A comma usually goes before those conjunctions, almost never after.
The only authority I've seen that supports So-comma is The Elements of Style, published 1918, which does not comment, but uses So-comma in 2 on-point instances and one off-point instance. The other 9 authorities I examined overwhelmingly (but not entirely) avoid placing comma after "so" when it means therefore at the start of a sentence or independent clause.
A few days ago, I combed through The Prince of Tides, War and Remembrance, Sophie's Choice, Light in August, Portnoy's Complaint, The Bluest Eye, Alias Grace, The Magus, The French Lieutenant's Woman, and Atonement — somewhere between 1.5 million and 2 million words. In all of that text, professionally edited and published by major houses, there were only 9 instances of So-comma in the sense of therefore, where it did not introduce a subordinate/parenthetical clause or phrase. In fact, even when introducing a comma-parenthetical, those writers (or editors) usually deleted the first of the two commas that would delimit it if it appeared later in the sentence. (@Cerberus_Reinstate_Monica states and illustrates that well in an earlier answer.)
Compared to those 9 instances of So-comma, there were (conservatively counted) 871 instances of So-without-comma in the usual sense of Therefore, plus hundreds of others in senses like Thus and To such an extent.
Of those 871 instances, 465 were at the beginning of a sentence; 51 immediately followed a semicolon; and 355 immediately followed a comma.
Link to Full Tabulation (PDF)
Although in my experience, So-comma was sometimes seen earlier than 2000 in under-edited business writing, it is mostly a child of the 2000s. The textual analysis overwhelmingly supports my lifelong impression that professional editors and writers throughout the 1900s nearly always avoided comma after so-as-therefore. That's why it sounds and looks strange to those of us of a certain age and breadth of reading.
That said, there are certain occasions when So-comma is needed for emphasis or to avoid ambiguity. ("So, long before the deadline, we began wrapping up the project" — to avoid "So long" or "So long before.")
But in Grammatically Correct, author Anne Stilman points out regarding emphasis that when speaking, we often pause or change intonation in places we don't punctuate with a comma. To create emphasis, Stilman recommends instead choosing one of italics, ellipsis or a dash to convey a pause or stress. I recommend the same when I edit fiction, where such emphasis normally occurs inside dialogue. The comma looks too accidental and unpolished.
So again, the best simple rule-of-thumb is to avoid comma-after-so (indeed comma after any FANBOYS) at the beginning of a sentence, immediately following a semicolon, or immediately following a comma. That will nearly always align you with great writers and editors.
