After Understanding English Grammar by Martha Kolln, grammar.ccc.commnet.edu ... advises the following:
Commas and Introductory Elements
- When a sentence begins with an adverbial clause, put a comma after it.
Although we had reviewed the film twice before, we never noticed these
details about the shooting.
As the day drew to a smoky end, the firefighters put out the last of
[To go beyond the basic improvements that we have already started work on, we will need to raise more funds.]
- It is permissible, even commonplace, to omit a comma after most brief introductory elements — a prepositional phrase, an adverb, or a noun phrase:
Yesterday afternoon we sat around waiting for Bill to arrive.
By evening we had become impatient.
Jauntily he walked into the hall.
[To go further we will need to raise more funds.]
- When a prepositional phrase expands to more than three words, say, or becomes connected to yet another prepositional phrase, the use of a comma will depend on the writer's sense of the rhythm and flow of the sentence.
After his nap Figueroa felt better.
After his long nap in the backyard hammock, Figueroa felt better.
- When an introductory adverbial element seems to modify the entire sentence and not just the verb or some single element in the rest of the sentence [ie this is a pragmatic marker rather than a true adverbial function] put a comma after it.
Fortunately, no one in the bridal party was in that car.
Sadly, the old church was completely destroyed.
On the other hand, someone obviously was badly injured.
[To be honest, I think that the Government has relaxed lockdown regulations too early.]
- Don't allow a brief introductory element to merge with something following it in a way that can confuse your reader. Try reading the following sentences without their commas:
Until the spring course lists will not be published.
Until the spring, course lists will not be published.
Inside the gym was brightly lighted and clean.
Inside, the gym was brightly lighted and clean.
- When a sentence begins with an Absolute Phrase or an adverbial Infinitive Phrase, put a comma after it. (If the infinitive phrase is acting as a noun and is the subject of the sentence, be careful not to put a comma between the subject and its verb: "To believe in one's self is a good thing.")
Their headpieces flapping wildly about their ears, the priestesses
began their eerie chant.
To escape with our lives, we would have to run for the exits.