「Aついでに、B」: A is the main action and B is another action done while peforming A.
V(dict. form)・(plain past form) + ついでに N + の + ついでに
The connection to "two birds with one stone" seems a bit on the weak side to me, but not entirely absent. I think the two birds saying has more to do with deliberately combining two tasks, while ついでに makes the second one more of an afterthought. I'd explain AついでにB as, A is the main thing I'm doing, and B wasn't necessarily even in my plans, but doing A makes it convenient to do B, so I might as well do B too.
Is the nuance a bit like the English expression: "killing 2 birds with 1 stone"?
I have heard as a standalone expression "あとついでに、〜する". "Oh, while I'm here I'll do 〜". If example, if you are in a convenience store and you suddenly remember to buy mints. "Ah, while I'm here, I'll buy some mints"
can you use this as a standalone casual expression to mean "while you're at it...?"
this page [simultaneous-actions-group] tries to explain some subtleties of nuance...
After doing some Googling, I think it means something close to "While you're at it", but much more flexable (ie, not limited to present or "you"). I saw it at the beginning of a sentance. The person requested that someone teach him how a broadcasting system of some sort worked, and the sentance ended. Then the next began with "序でに," and it was requested that a related concept also be explained.
I might have thought this was usually used for past things, but it can also be used for the future. パリへ行くついでに、ロンドンに寄る Since I am going to go to Paris, I will also stop by London
V-plain or V-plainpast + ついでに
ex#1006 From the content, wouldn't it be better to say, 仕事に行く時に、ついでに郵便局で手紙をだした。?