How about Senator Joe?

The pardon removes any legal impediment to his candidacy, so yeah, Apraio's free to run. Whether his past legal troubles represents any sort of political baggage for Arpaio is a harder question to answer. The Arizona Republican Party is going to view him as a potential Roy Moore 2.0, not in terms of the pedophilia - there's no hint of anything along those lines, at least not at the moment - but in terms of his being more a right-wing populist than a serious thinker on policy. Congresswoman Martha McSally is the establishment's choice to take the nomination and, they hope, succeed Flake in the Senate. Conservative gadfly Kelli Ward tried unsuccessfully to run against McCain and would probably have run for this seat even if Flake had sought re-election. So at the moment you've got a existing office holder favored by the establishment, a perennial gadfly candidate and a celebrity populist all vying for the nomination. Who would you put your money on? From the outside your instinct is to say the guy with the name, which means Arpaio. But it's not as simple as that. Arizona knows Arpaio's record as sheriff better than the rest of the nation, and attitudes about him are not universally positive - including among conservatives. He's widely seen as a rogue sheriff who became a law unto himself, but couldn't be touched because his popularity in Maricopa County allowed him to keep getting re-elected - until 2016, when Maricopa County voters seems to have had enough of Arpaio fatigue and actually elected a Democrat sheriff rather than giving Sheriff Joe another term. By the way, for what it's worth, his federal conviction was a sham. The federal government ordered him to stop targeting illegal aliens in traffic stops, contending that this amounted to racial profiling. That's absurd. Since illegal aliens are by definition foreign, you can possibly target them without there being some sort of racial element to what you're doing? Sheriff Joe was in the wrong legally to ignore the order, but the order itself should never have been issued. The whole thing was a disgrace. He's not Roy Moore, but when you see establishment Arizona Republicans banging their heads on their desks today, understand why. A Sheriff Joe candidacy is not what any of them want.