Concerning CIWSs and missiles
Scenario
A ship with advanced radars will probably detect an incoming LO surface skimmer at around 10km. This means that, in the case of a missile travelling near mach1, the time from detection to impact is around 30s. Likely there will be multiple missiles performing complex evasive manouverings.
Upon detection, the defensive system of the ship will have to analyse the threath, identify it (you don't want to shoot your own assets) and track it before a missile can be fired.
I don't know how long this response time is for a modern systems, but it's probably not insignificant. Let's for the sake of fun say that this response time is 3 secs.
When the AD missile is fired it will have to travel to it's target. Let's say that the AD missile travels at an average speed of Mach3 (some 900-1000 m/s, I know modern missiles are faster, though they have to accelerate and they don't follow a straight path).
Then, from the top of my head, engagement happens after 7-8secs after the AD missile is fired, at a distance of some 7km from the ship.
Let's assume that we have a miss, The ship will have to spend some time detecting the miss and that a missile is still inbound, before it can shoot again. Let's assume that this delay is 3 secs.
Then the 2nd missile is fired while the anti ship missile is some 6km away. What then happens depends on the missiles initial flight charracteristics of the AD missile - which I don't know, at best maybe the missile could be able to engage it's target at some 4km distance from the ship or some 13 secs before impact. Though I don't think that's realistic. a vertical launched missile will follow a far from straight path, also is the illuminater agile enough to track a fast moving target that close?
In any case, the second shoot, won't leave much room for error/delays or unfavorable corrections to my amature assumptions on speed and response time of the defense system.
Likely, in any case the point of engagement of the second shoot will happen within or near the engagement range of a modern 30-40mm gun-CIWS. Though it should be so, that you will have one clean shoot with a missile on a horison detection of an inbound missile, it appears to me as likely that the second shoot can might as well be done with a gun.
Some CIWS systems can fire AHEAD ammounitions (Like the Orlikon contraves CIWS gun on the absalon class). As far as I remember this weapon fires some 800 30mm shells a minute. And during the (very short) time that a given shell is in the barrel, the weapon can actually "time" (set the fuse?) the shell, so that it will explode at a predetermined optimal distance. Each shell explodes in 150 tungsten fragments each at some 3-4g and the relative speed of impact on it's target will likely be some 3-4mach or the same as a high powered rifle.
So in a 10 sec burst of fire (about the time it will take the anti ship missile to close the distance to the ship, after being in range of the gun) this weapon will fire some 140 shells laying out a dense "cloud" of some 20000 high speed metal fragments, in a shotgun like effect.
As far as I know the 76mm OTO Melara Super Rapid, can also be equipped with this type of ammunition. this gun will be able to initiate engagement at horison. And eventhough it has a slower rate of fire, it's shells are larger, carring many more fragments, As far as I know the ready to fire magazine is around 80 rounds, the gun should be able to fire 60 rounds in 30 secs (the time between detection and "game over").