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It's the same hat and it travels once.

That depends on whether he puts the stolen hat on the front or the back

The best way to do time travel is to put characters and entertainment first. You could give the illusion of consistency like in Back to the Future, where most people won't notice the problems. So long as most of the audience buys it, you don't need bulletproof logic, just good enough logic.

At the same time, you can have Dr. Who logic where they admit things don't make total sense, but have general rules that make obstacles for the characters when convenient. The rules work best when they challenge the characters and aren't relied on to help them. Try to maintain reasonable consistency over a short stretch of time, even if it contradicts lore from years ago.

The most extreme types are cartoon logic and total consistency. Cartoon logic allows the writer to do whatever is funniest or the most entertaining. In many cartoons, other laws of physics are broken for laughs, so time being broken isn't a huge leap. In totally consistent stories, time travel usually harms the characters the most, and every action has consequences. No time stabilizing watches or other stuff. This is a headache to write or watch, so few stories are actually like this.