Right click the toolbar and select Customize. Add or remove buttons from the dialog box that opens or move the dialog box out of the way and drag and drop the buttons where you want them.

This customizes the toolbar. It does not bring up the buttons on the top right of the opened message, such as "quick reply" etc. ending with "More". That is the row of buttons I want to change.

Maybe you should post a screen shot of the buttons you are working with. This method works with both the Mail Toolbar and the header bar buttons on a message tab and the Message Pane. Of course you have to right click the one you want to modify to get the desired results. I am still using ver. 31.6 due to the issues with ver. 38 but I would be surprised if they changed this operation since it has worked this way forever.

and that row of button is a toolbar therefore if you right click it you can customize it just like Airmail said.