set one's cap for, to

Pursue someone romantically, as in We all thought Anne had set her cap for Joe, but we were wrong. In the 1700s this term, which may have alluded to donning one's best headgear, was applied to members of either sex, but by the early 1800s it generally described a woman chasing a man. It is probably obsolescent.

To attempt to attract and win as a mate.

To pursue someone as a potential mate. This term dates from the eighteenth century, and although at least one writer believes it refers to ladies choosing their most becoming headgear in order to attract gentlemen, it was originally applied to both sexes. By the early nineteenth century, however, it was used mostly for females chasing males, as in Byron’s Don Juan of 1832 (“Some who once set their caps at cautious dukes”) and Thackeray’s Vanity Fair of 1848 (“Have a care, Joe; that girl is setting her cap at you”). Shirlee Emmons’s biography of Lauritz Melchior (Tristanissimo, 1990) says Melchior’s children believed “that Kleinchen deliberately set her cap for this young man who lived alone and far from his family.”