“The Chaser” is a great story. As your question suggests, the title seems a bit odd because the story never specifically mentions a “chaser”; however, I do see the title as quite appropriate for three reasons.
First, I believe that Alan Austen is very much chasing Diana. She is the woman that he desires more than any other woman.
“She will, when she has taken this. She will care intensely. You will be her sole interest in life."
"Wonderful!" cried Alan.
The problem is that Diana doesn’t desire Alan the way that he desires her. If she did, he wouldn’t be needing a love potion in the first place. While the story doesn’t give specifics about any of Alan’s previous attempts to woo Diana, I believe that he has made efforts to woo her. Those efforts have failed, and now Alan is chasing another tactic to win Diana’s heart. In this way, Alan is the chaser.
The second “chaser” in the story is Diana herself. I suppose, to be more specific, Diana would be the chaser. Once she drinks the love potion, she will want nothing but to pursue Alan’s affections. She will want to know what he’s thinking, where he’s been, and where he’s going.
"She will want to know all you do," said the old man. "All that has happened to you during the day. Every word of it. She will want to know what you are thinking about, why you smile suddenly, why you are looking sad."
Diana will be so busy chasing Alan and demanding his attention that she will be unable to focus on anything else.
“You will be her sole interest in life."
The final “chaser” in the story is the $5,000 poison that the old man in the story calls a “life-cleaner.” He describes the poison as highly effective and completely untraceable.
"Here is a liquid as colorless as water, almost tasteless, quite imperceptible in coffee, wine, or any other beverage. It is also quite imperceptible to any known method of autopsy."
"Do you mean it is a poison?" cried Alan, very much horrified.
"Call it a glove-cleaner if you like," said the old man indifferently. "Maybe it will clean gloves. I have never tried. One might call it a life-cleaner. Lives need cleaning sometimes."
The old man spends a fair amount of time hinting that Alan will one day need to return for the poison. Alan will need it to kill Diana and free himself from her affections. If that were the case, the “life-cleaner” would be the second drink that Alan gives to Diana. A second drink that quickly follows the previous alcoholic drink is sometimes referred to as “a chaser.” A chaser is a drink that is typically a little more palatable than a shot of straight liquor. It is used to help wash down the shot. Alan’s second drugged drink to Diana is her chaser drink. I suppose that if Alan were to actually need to use the chaser, it’s because he’s trying to make Diana more palatable again . . . by making her dead.