Sinatra recorded three albums for Capitol with Billy May at this time, the swinging trio "Come Fly with Me"; "Come Dance with Me"; and "Come Swing with Me." It's clear that Sinatra and May are getting a little tired of each other by the third collaboration. Check out May's later, rather snide version of :"Stranger's in the Night": https://youtu.be/3mbnLOzgKRw?si=532GKr2bPxqHjuJc
Sinatra also recorded two post-Riddle albums of torch songs with Gordon Jenkins: "Where Are You" and "No One Cares." Not quite up to the standards of the Nelson Riddle songs-for-suicide albums of the 50's but quite good.
This is a fine perspective on an important era for Sinatra and American music in general.