My grown son and ex-husband went in 2008. My trip was a long time ago 1983.
There are three bridges - northern part of Israel, the middle and the southern part. Which area will you be in?
1983 -We traveled to Israel from Jordan and back to Jordan again via the Hussein bridge. We spent 5 days in Israel. Since our travel originated in Jordan, we had to get a re-entry visa from the Jordanian Ministry of the Interior to cross the Hussein Bridge, since we hadn't made prior arrangements. We went by bus to the bridge (going and coming) and took a taxi back to Amman, sharing with others. Only certain taxis have permits to go past a check-point.
2008 - My son and ex got visas to enter Jordan/Israel before leaving the US, but my ex thinks you may be able to do this at the bridge (double-check before you leave the US).
They took the northern bridge from Israel to Jordan, going to/from by bus. Coming back, they took the Hussein bridge. They had a driver in Jordan and he couldn't go past the check-point, but there was a shuttle to the bridge. They took a bus from the bridge to Jerusalem.
If you go to Jordan, I hope you'll take the time to go to Petra. It's amazing! They tell me it isn't the remote place it was when I went in 1983. Back then, we had to stay at the Government Rest House. Now, there are hotels.
Jerash is a great excursion out of Amman, if you like Roman ruins.
Cameron