序（ついで）に is a 副詞（ふくし） (adverb) in Japanese that is used in a similar way to ‘while you’re at it’, or ‘on the occasion of (A)’ in English. 序（ついで） itself is a noun that means ‘precedence’ (in terms of being the first thing to happen). Due to this, the literal meaning of 序（ついで）に may be thought of as ‘with (A) as the precedence, (B)’.
序（ついで）に is commonly used with verbs, or nouns. In the case of nouns, it will need to be preceded by the 格助詞（かくじょし） (case marking particle) の.
-
買（か）い物（もの）に行（い）くついでに、郵便局（ゆうびんきょく）に行（い）ってこの荷物（にもつ）を出（だ）しておいてくれる？
Can you bring these packages to the post office while you go shopping?
散歩（さんぽ）のついでに寄（よ）って行（い）ってください。
Please come visit while you are on a walk.
お母（かあ）さんの所（ところ）にテレビを返（かえ）しに行（い）くのでしょう？序（ついで）にこの、私（わたし）が作（つく）った梅酒（うめしゅ）を持（も）って行（い）って。
You are going to go to your mothers place to give back the TV you borrowed, right? While you're at it, can you take the Umeshu I made?
Just like ‘precedence’ in English, 序（ついで）に implies that (A) is the instigator for (B). Because of this, 序（ついで）に is a good way to convey that something is not urgent, but, if (A) if happening, it is the perfect chance to do (B) as well.
序（ついで）に is quite a bit more formal than
間（あいだ）に or 時（とき）に, and will usually only be used for ‘one-off’ things, rather than regular occurrences.
-
友達（ともだち）を家（うち）に送（おく）ったついでに、床屋（とこや）に行（い）った。 (One-off occurrence)
On the way back from driving my friend home, I went to the barber.
旦那（だんな）はいつも、私（わたし）が寝（ね）てる間（あいだ）に帰（かえ）ってくる。 (Regular occurrence)
My husband always comes back while I am sleeping.
私（わたし）は晴（は）れているときに出（で）かける。 (Regular occurrence)
I go out when it is sunny.