As I understand it, the Allies had air supremacy more or less throughout the Battle of Sicily.
The accounts I’ve read indicate that surrounding the island by occupying or interdicting the Straits just seemed to slip everybody’s mind till pretty late in the battle.
http://www.airpower.maxwell.af.mil/airchronicles/cc/renner.html
A LOT of Axis forces got away, somewhere in the vicinity of 125,000.
Most attacks on the area was directed at the port facilities and unfortunately they did not concentrate on the ships ferrying troops off the island.
The biggest failing by the Allies on Sicily though was the AFHQ's refusal to believe the Germans were evacuating the island at all. As late as Auguast 12th, Alexander had reported to Sir Alan Brooke that he had no indication that the Germans were attempting to leave Sicily. This was dispite the 8th and 7th Army's constant reports to the contrary along with supporting reconnaissance photography, reports from captured POWs (Italian and German) and captured evacuation plans of the Hermann Goring Division which showed the planned evacuation.
All this evidence showing that the Germans were bugging out and yet on August 14th, AFHQ's G-2 division issued a report that stated "there is no evidence of any large scale withdrawal of troops from Sicily."
A good author to follow if you want to read more on Sicily is Carlo D'Este. Along with numerous journal articles on the subject he also wrote Bitter Victory which covers the Sicily campaign.