I use a simple, generic leather passport wallet to protect the cover. Due to the reason stated above about the immigration officers kept removing (no tossing yet) the passport from the wallet to scan or to stamp, I would now first withdraw the passport out of the wallet before approaching the immigration counter.

Also, to save the officer from fanning the passport front and back looking for the picture, I would mark this page, and place the opened passport at the counter with my picture staring right at him/her.

It also projects goodwill to greet and thank the officer with the language of the country you are visiting. So far I have never failed to get a smile and/or equally pleasant greetings in return.