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

try to attract as a suitor. dated