He was listed as crew on the 1921 voyage (also he is probably the Philip Coole, no middle name, also 16, arriving in 1920 - also crew). So I don't think he stayed in the US unless he jumped ship for a little while - and you may have trouble finding his entry into Australia if he was crew, too.
Australia National Archives recordsearch shows that he did enlist for WWII in Australia, if that helps any.
The National Archives in Kew has the The Fourth Register of Seamen, 1918-1941, on which he may be listed. (You could also contact Southampton Archives, who hold the originals).