Sounds like a good book! History is not a straight line, and there is no single reason for any policy. However, I spent six years in Europe during the cold war in the Air Force and Army respectively. The Air Force period began with the Hungarian uprising and ended with Eisenhower sending the Marines into Lebanon. The major events in the army period were the U2 incident and the rise of the Berlin Wall. I have a surface knowledge of events in Europe from 1956 through 1962.
Because of the coups in Iran and Guatemala, I was never a big fan of John Foster Dulles, but both he and Eisenhower were very experienced in foreign policy. Dulles had been around since Versailles, and Eisenhower had to deal with all the European leaders as Supreme Commander in Europe during WWII.
For the same reasons I didn't like their coups in Iran and Guatemala, I liked their support for Egypt during the Suez Crisis. One reason Eisenhower might not have been too happy with Israel is that they had run a disinformation campaign, attempting to blame Egypt for arson attacks against US and British government offices, along with cinemas showing films from those countries. A Labor government was in power in England during this period. It was reported in the papers at that time, but Avi Shalaim's excellent book The Iron Wall, Israel and the Arab World also mentions it.
During an interview with a Palestinian diplomat, he said, as a child, there used to be baseball-type cards with Eisenhower's picture that were very popular. He had a good record in the Arab Middle East. American Foreign Policy in the Middle East during this period must be viewed though the prism of the cold war.