World War I prevented Salmond from developing his international career further at that time, but he resumed building a reputation in chamber musicafter the war. His performances in this period included the premieres, on 21 May 1919, of Edward Elgar's String Quartet in E minor and Piano Quintet in A minor at the Wigmore Hall. After Salmond's performance of his quartet, Elgar entrusted Salmond with the solo part for the debut of his most personal and heartfelt work, his Cello Concerto in E minor with the London Symphony Orchestra at the Queen's Hall. The premiere, on 26 October 1919, proved to be a disaster. The performance was scheduled such that Albert Coates, the conductor of the London Symphony, would conduct the rest of the programme and Elgar himself would conduct the concerto. Coates, a self-important man, was well known for using up to forty-five minutes of his hour of rehearsal time lecturing his players. After Coates consumed an hour of Elgar's rehearsal time, Elgar—who was until that time waiting offstage for his chance to rehearse—uncharacteristically exploded with anger. The severely under-rehearsed performance which followed received scathing reviews, with Ernest Newman stating that "the orchestra made a public exhibition of its miserable self". Elgar later said that if it weren't for Salmond's diligent work in preparing the piece, he would have pulled it from the concert entirely. On 10 November 1920, the inaugural concert of the City of Birmingham Orchestra was given at Birmingham Town Hall, with Elgar conducting a concert of his own works, including the first performance of his Cello Concerto in that city, with Salmond as the soloist.
On 29 March 1922, he made his American solo debut in New York at the Aeolian Hall. He settled in America, although he returned to England and Europe for tours. He was appointed to the Juilliard School's faculty in 1924, and became head of the cello faculty at the Curtis Institute of Music a year later—a position which he kept until 1942. However, still bruised by the experience of the first performance of the Elgar concerto, he did not teach it or play it outside England. He was highly regarded in America as a teacher, with pupils including Orlando Cole, Suzette Forgues Halasz, Bernard Greenhouse, Leonard Rose, Daniel Saidenberg, and Alan Shulman. He also received great appreciation as a performer. In 1924, he appeared at Carnegie Hall in a well-reviewed piano trio with pianist Ignacy Jan Paderewski and violinist Efrem Zimbalist. He had a broad taste in music for the cello, including works by contemporary composers such as Samuel Barber, Ernest Bloch and George Enescu. He last returned to England in 1947; he died in New York.