Any proofreader, professional or not, knows that the second readthrough is super important. You’ve done the first one, fixed all the obvious mistakes, devoured the accuracies the way a lioness devours a zebra, and then you’re suddenly at the end of your document. But your editing doesn’t end there.
The second readthrough is something everyone should do with their own documents, whether it’s someone else’s work you’re proofreading, or your own. If you don’t read your own work, how are you going to figure out if there’s anything wrong with what you put? I don’t notice my own typos on the first time around myself. You can use Word’s editor if you’re writing on Word, which will catch some errors, but it’s important to find and catch all the errors they miss yourself.
And trust me, you will make errors, no matter how good your content is. Everyone does, it’s no reflection on you. Go ahead, embrace it! I make mistakes, you make mistakes, we all make mistakes. It’s why the second readthrough is so important. Maybe you’ll be lucky. Maybe you’ll have found all your mistakes already. But usually, the second readthrough will help you find just one thing you’ve misssed.