This was all set up earlier in the movie. Sharla is looking through the pictures at the pizza place she works while on the phone with Rex. You know it's him because she says "Don't worry, it's not like you can see your face" indicating Rex is nervous someone is going to find out about their affair.
Joe also has several days (at least one day planned out) when he is to be alone with Dottie at their trailer, giving him hours upon hours to possible go through their belongings and find wherever Sharla had hid them.
And he didn't get the pictures from her phone, they were in a small, red photo binder like what some people keep coupons in which explains why she probably wouldn't have carried them on her at all times, unlike her phone which Joe wouldn't have had easy access to.
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