28 January 2016: Corrections and additions.
At the base of the leading edge of the vertical fin of the F4H/F-4 Phantom, there is a ram air inlet.
The best guess now is that the air was piped down to the interior of what appears to be double-walled side panels aft of the engine tailpipes and exited through a large slot on either side of the aft end of the tailhook.
Still unanswered is why the configuration of the inlet changed over time.
For the F-4A/B/C and probably the D, it was a simple inlet almost flush with the top of the fuselage as shown in the first illustration with what appears to be a very shallow boundary-layer-removal slot at the bottom of the opening that dumps the slow moving air into the aft fuselage. (This is F-4A BuNo 145310 currently being refurbished for flight.)
Craig Kaston Photo
But wait, there's more: On the K/M, the Spey-powered Phantoms, the lower side of the inlet curves inboard so that the horizontal splitter extends slightly outboard of it for a short distance aft of the inlet.
That little exterior curve does more closely match the shape of the internal ducting. (Note that the internal louvers have disappeared with the introduction of the horizontal splitter.)
So: Why the ongoing "improvements"? There couldn't have been a great deal of benefit because it appears that the Bs were never retrofitted with the later inlet, not even on the Bee Line when they became Ns. It's possible that with the introduction of the Spey, the cooling being provided was marginal, necessitating a tweak in the inlet to reduce duct losses.