The song from A Hole in the Head that won the 1960 Oscar for Best Original Song was “High Hopes.”
The song was written by Jimmy Van Heusen, with lyrics by Sammy Cahn, for the 1959 Frank Sinatra film A Hole in the Head. Sinatra plays Tony Manetta, a widowed hotel owner in Miami who struggles to keep his business afloat while relatives pressure him to sell. The song’s optimistic theme fits the film’s hope that Tony can overcome his problems.
“High Hopes” won at the 32nd Academy Awards, held in 1960, which honored movies released during 1959. The award recognized the songwriters rather than simply Sinatra’s performance. Sinatra’s recording became one of his signature interpretations, and he later used the song in political campaign contexts, helping give it a life beyond the film.
The neighboring Sinatra songs in this question come from different films: “All the Way” was introduced in The Joker Is Wild, while “Young at Heart” and “Love and Marriage” are associated with other recordings and productions. The Oscar connection specifically points to “High Hopes.”