set (one's) cap at (someone)

dated To try to attract, secure, or win someone as a romantic partner or spouse. Said especially (though not exclusively) of a woman in pursuit of a bachelor. Well, if he insists on remaining so inhospitable, then I shall simply set my cap at a man with a greater sense of charm and decency. The ladies of this town shall all be setting their caps at Mr. Rutherford, now that his inheritance has left him quite wealthy. But don't you find it rather unseemly for a man of his age and station to set his cap at a girl who's barely of voting age?

set your cap at someone

BRITISH, OLD-FASHIONED

If a woman sets her cap at a man, she tries to make him notice her, usually because she wants to marry him. If I were a little younger, I might have set my cap at him myself. Note: The idea behind this expression is that in the past women would wear their best cap in order to attract the attention of a man they wished to marry.