In OTL, the British decision to withdraw implicitly meant that France had to withdraw as well since this was a British eld operation and air superiority depended on the use of British airfields in Cyprus. What forced Britain to agree to a pullout was US refusal to provide loans or oil and Britain at the time couldn't get any oil from the Middle East because Nasser had blocked the canal.
In light of the above, what if Eden ignores US pressure and has Anglo - French forces push on and capture the Canal. The task force can then clear it for British shipping to pass through. This means Middle Eastern oil can flow through the canal to Britain and ease pressure on the pound. I'm aware that some of the Arab oil states had imposed an oil embargo on Britain, however, not all of them had so in the event of a British recapture of the canal there would be oil coming through.