I’ve never heard this as a closing. I’ve heard it used, rarely, in conversation but I think of it more as a word from the Great Gatsby era.
Most letter closings say something about the relationship between the writer and the reader. “Best regards”, “Sincerely”, “See you soon”, “Thanks”, and “Thanks for your time” are all saying something from one person to the other.
“Swimmingly” just seems to say “I’m doing great!”
FORMAL A signature or short piece of writing at the end of a document.
“He signed the letter and added a subscription”
We close a letter with words/phrases like “Thanks, Thanks and regards, Sincerely yours, Yours faithfully” or the like. We call it subscription or concluding words.
I think that’s a very formal or even archaic use of the word. In most cases, a subscription is an arrangement to receive something, typically a publication, regularly by paying in advance.