How is my sentence? 我以为我没有工商卡，可是，我顺便找到了。 I wrote this message to a friend on WeChat, who previously helped me open a 工商银行 account:


  我以为我没有工商卡，可是，我顺便找到了。我说错了。
  I thought I had no ICBC bankcard, but, I stumbled upon one.  I said the wrong thing.


I was doing something else, and I found another bankcard from 工商银行 which I had long forgotten about, and realized I told him the wrong thing earlier.

I'm a bit uncomfortable with what I wrote:


I'm not sure if 工商卡 is valid; maybe I should have written 工商银行卡 but it's a bit long.  It's not a credit card, but just a simple bankcard, so I didn't want to use 信用卡.
This is my first attempt at using 顺便 in a sentence.  I think it's okay, but I'm not 100% sure.  I don't see it violating advice given here What is the difference between 随手 and 顺便？


I'm just after a critique of this snippet.
It sounds weird using 顺便 here. I think you should use 无意中 (by accident/undesignedly) instead, e.g. 我以为我没有工商卡，可是（or use 结果 instead to emphasize the opposite result）我无意中找出来一张。

Note that 顺便 is intentional. e.g. 今天我去见我导师，顺便去大学图书馆借本书。The action that going to the library by the way is still intentional, it's planned. But for this case, the result that finding out the card is unmeant, undesigned. As you said 我以为我没有工商卡, so you won't try to search the card intentionally.

PS: 工商卡 is valid, but we use 工行卡 or 工行的卡 more because 工商银行 is abbreviated to 工行 ususally. Similarly, you can say 招行/建行/中行/农行 etc.