Scottie does a pretty poor job of following Madeleine, not actually being a private eye. He's often plunging directly into areas where the two of them are conspicuously alone, such as in the alley outside the flower shop and the turnaround below the bridge. How Madeleine fails to notice that she's being followed seems like movie convention, but it turns out that she wanted to be followed and was only pretending not to notice him.
Not only that, Scottie wasn't a private eye. He was a police detective. He's used to going to places, asking questions. And since he retired, it's been some time that he probably isn't as good at tailing someone as he used to be. Scottie even says "This isn't my line" when Gavin refuses Scottie's recommendation for a private eye firm. So not only did Madeleine want Scottie to follow her, Scottie wasn't good at it due to the circumstances.
A bit of Fridge Brilliance that not many people noticed when it comes to then, but is more noticeable now thanks to various shows about crime scene investigations, that something about seeing Madeleine's body on the roof after her "suicide" that seems off. When Scottie looks down at it, her body is seen pretty close to the base of the tower when it landed on the roof. If she had jumped similar to how we see "Madeleine" jump into the bay, the body would be farther away from the base of the tower. Then we're shown in Judy's flashback that Gavin had tossed the real Madeleine off the tower, which explains why the body impacted closer to the tower than farther away if she had jumped: she dropped down instead of out.
Many may not notice this, but when Madeleine goes to Scottie's apartment to drop off the letter has a bit of foreshadowing fridge brilliance. When she talks with him, she says, "I didn't know the address, but I had a landmark. I remembered Coit Tower, it lead me straight to you." The brilliance is that it's a foreshadowing when you realize that she used a tower to reach her end, much like the tower that lead to Madeleine and Judy's actual end.
Outside of the Troper Name with the "how'd she get out of the hotel" scene, there's another one with that exists, but with Midge. If Midge is in love with Scottie and is jealous that he's fallen in love with another woman, why did she call off the engagement when they were in college? We never get an explanation behind why she did, and all we know is that she called it off.