Well,
before Constance the very first Attic Bride at Disneyland (Who was simply called "Beating Heart") was tucked away in the corner on the left side of the ride track, right by the bend where the Doombuggies make that hard right turn - today that spot is basically between the 2nd and 3rd Husband Portraits you see in the Attic (Frank and Reginald). This photo is from Disney
World's Attic, not Disneyland's, but you can imagine the first Bride would have been basically right in front of that Tapestry:
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After The Hatbox Ghost was removed from the ride the Bride was moved to his spot by the exit so that the Attic still had some sort of finale element to it. But she was on the Right side of the track, not the Left where Constance is now. The Bride jumped from the right side of the track to the left in 2006 when Constance was added because they didn't want to close the ride for her install, so the Old Bride stayed in her spot on the right while they somewhat secretly installed the one on the Left.
Here's a Blueprint of the Attic with all the new/old Bride and Hatbox Ghost spots marked out. The new Hatbox Ghost's placement is just out of frame on the bottom, since they moved him onto the balcony overlooking the Graveyard instead of putting him back where he started:
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Mind you, all this is Disneyland-centric - at Walt Disney World the Bride was always at the end of the Attic on the right side until 2007 when Constance was added, and she's always been in the same spot as DL's Constance at the end of the Attic on the left side of the track. Meanwhile, The Hatbox Ghost has never been installed at Walt Disney World . . . yet!