Jesus observed the Passover of the Jews somewhere in the upper city of Jerusalem, the present Mount Zion. Scholarly consensus locates the house of Caiaphas at a site near the Cenacle on Mount Zion. Another site, further down the slope toward the Pool of Siloam, is held by some as the house of Caiphas. It is called St. Peter in Gallicantu (cock-crow). According to this tradition, it is the place where Jesus was taken after His arrest (Matthew 26:57). If correct, this would be the site of Peter’s denial of the Lord.
The late W. Harold Mare says,
Whatever interpretation is accepted, one is impressed with the ancient stone stairs that run from the vicinity of the Cenacle down past the Church of St. Peter in Gallicantu to the Pool of Siloam, stairs that are thought to be from the Jewish period. It could be that these were the very stairs used by Jesus as he went down from the Last Supper to the Kidron Valley and on to Gethsemane (John 18:1). (The Archaeology of the Jerusalem Area. Baker, 1987)
My understanding is that Jesus would have observed the Passover and instituted the Lord’s Supper in the Upper City. He then went with His disciples to Gethsemane. From there He was taken to the House of Caiaphas. Jesus may have taken these steps in both directions. Here is a photo of those stone stairs that I made last year.