Most airlines eliminate cheaper seats as you get closer to departure. The vast majority of last minute bookings are business travelers who will pay whatever it takes to get there when they need to. So no real motivation to sell discounted seats (especially since last minute for business needs can truly be last minute).
The route you are looking, USA to southern Europe, will be busy in spring, so the cheapest seats are likely selling well already. You could try a broad search, say til 1 June, find the cheapest ticket and then try to standby for an earlier departure. Or two, just bite the bullet and book the best fare for the dates you want to go. But "standby fares" went the way of the dinosaurs with paper tickets ;-)