What the Official History calls "the second phase" took place concurrently with the abortive Operation Battleaxe, from 14 June to 22 June 1941. When the Vichy French forces withdrew from Damascus, the nature of the bombing campaign changed. Now, the Blenheims were able to attack "targets of opportunity" on the roads. About half of the bombing effort in this phase was devoted to hitting moving enemy forces. The fighter forces were still dedicated to providing air cover for the warships, in this period, bringing them into contact with German aircraft, such as Ju-88's in the anti-shipping role. The Vichy air force was gradually being written off, with few British losses.
We also knew that the large destroyer Chevalier-Paul, just arriving from metropolitan France, was sunk by air attack. There is Chapter 8 from The Royal New Zealand Navy online. This Chapter is called "Operations off the Coast of Syria". The book confirms that the aircraft that sank the Vichy destroyer Chevalier-Paul were six FAA Swordfish flying from Cypress. This is largely based on the account in Vol.II of the Official History.