Daniel, I can't answer you definitively, but I'm pretty sure you don't need a visa if you're just in transit, even though you're spending time landside.

I can't be definitive because we'd been in Jordan earlier that month and did have a Jordanian visa already in our passports.

The process is:
- after de-planing, go to the RJ Transfer Desk, airside.
- They'll give you a green transit card which you present to Passport Control.
- They stamp the card, and your passport, then you're free to go landside, but only to the hotel. RJ escorts you to the hotel shuttle.
- The hotel KEEPS the transit card until you check out, then returns it just before you get on the shuttle back to the airport.
- Back at AMM, you go through a special Passport line for in-transit travelers who are holding the green transit card.

My sense is that the transit card serves as a 24-hour visa. That said, if you're theoretically "at the hotel" for 23 hours, I don't see why you couldn't hire a cab or driver to take you into Amman. However, my preference would be to instead go to the Dead Sea for the morning, or up to Jerash. Jerash is about an hour north of Amman. There are some incredible roman ruins there, which take a couple hours to see.

The Dead See resorts are about an hour west of the airport. If you go to the Dead Sea, you might need to arrange beach access since they're mostly privately controlled by the hotels. But the hotels all seem to sell day passes for a fee, which includes a towel. Bring your own suit. Floating in the Dead Sea is a blast. Take a mud bath on the beach. We stayed at the Holiday Inn Resort Dead Sea, an absolutely beautiful hotel. It's also the closest to the airport.