[verb phrase 1],+ 顺便 + [verb phrase 2]
[subject] + 顺便 + [verb phrase]
顺便 marks an action you do because you're already doing something else – no extra trip, no extra effort. It goes before the verb of that second action.
The main action comes first, and 顺便 heads the second clause. If the second clause has its own subject, 顺便 follows it.
顺便 + 问 / 说 brings in a side question or remark: "by the way". It is polite, because it marks the question as minor.
"While you're at it" closes an English sentence. 顺便 has to sit before the verb it goes with.