I can think of a couple of other early jets before the F-4 that found use in both Navy and Air Force colors. The Douglas A3D/B-66 came about because of the Navy's desire for a nuclear strike jet aircraft. They put on a little show of force during the 1956 Oklahoma City airshow when a couple of them launched from somewhere in the Gulf of Mexico, I think, and did a sweep over Will Rogers airport. The AF used the B-66 mostly in recon mode, I believe.
The FJ spinoff of the North American F-86 was one of the first really good Navy jet fighters and served both Navy and Marine wings.
Then there was the F-111 which MacNamara tried to force down the Navy's throat. I have heard that if it was parked on the flight deck in the standard tail-outboard position, it was likely to fall off into the water. But they served the AF for years in various tactical roles.
Vought's F8U was certainly one of the best air superiority fighters of its day on land or sea, but the great F8U-3 follow-on lost out to the F-4 when the Navy opted for two engines/two seats. Then it gave birth to the A-7 ground support fighter for the AF (short little ugly feller).
Are the French still using the F-8 ?