At the conclusion of her 31st deterrent patrol, "Alexander Hamilton" returned to Charleston in November 1972 and in January 1973 began her second refueling overhaul, combined with a conversion to carry Poseidon missiles, at the shipyard of the Newport News Shipbuilding and Dry Dock Company. The work on those two modifications lasted for over two years. Upon its completion, she carried out shakedown in April 1975 and devoted the remainder of the year to training and various post-overhaul trials. She conducted her 32nd and 33rd deterrent patrols in the early part of 1976. "Alexander Hamilton" concluded the 33rd patrol at Holy Loch in May 1976 and conducted her next three deterrent patrols from that base. While on her 35th patrol, she visited Port Canaveral, Florida, and New London, Connecticut, before concluding that patrol at Charleston in March 1977. During March and April 1977, she completed refit and conducted refresher training. In July 1977, she departed Charleston for another deterrent patrol which ended with her arrival at Holy Loch in September 1977.