Let's see.
First off there's the five service branches, the United States Army, the United States Navy, the United States Marine Corps, the United States Air Force and the United States Coast Guard, (the National Guard and Air National Guard aren't really separate branches but reserves for the army and air force).
Well you can create a sixth branch, the United States Military Medical Corps, a separate medical service that provides field hospitals, shipboard medical care and battlefield medics to the other branches. There are a few countries in NATO that already have this arrangement and the USMC depends on the USN for it's medics and hospitals, and it's an idea with a great deal of merit as medical needs are pretty similar for all the services and the USMMC would allow you to combine training and funding into one service without the redundancy of four different ones.
You can also create a seventh branch as the OP suggests by creating an independent strategic missile force. Sometime in the 40's or 50's the administration gets tired of the tri-service fighting over the nuclear pie and worrying about having three different chains of command with nuclear release authority decides to centralise nuclear weapons under one command, the United States Strategic Weapons Force. USSWF takes over the USAF's missile silos and nuclear armed bombers, the USA's tactical nukes and puts Weapons Release Officers on the USN's missile subs.
Finally there's the possibility of an eighth branch. A gendarmerie. It's a bit tricky to pull off under the US system but it could be doable. An independent military police force for guarding military bases and providing law enforcement for all the other services. (The biggest TV show of this TL is Naval USGIS)
So that's now seven service arms.
Also as the OP suggest the USArmy can retain it's own air force. Close air support has always been controversial in the US, the army doesn't think the air force pays it enough attention and the air force thinks it distracts from their real job of flying supersonic jet fighters and really big bombers. So when the USAF is formed in 1947 it takes air defence, strategic bombing and airlift while the USAAC takes tactical and close air support and helicopters. After a few years the USAAC becomes the USAAF again and becomes a separate from the army much like the USMC is separate from the navy.
The USN keen to get into the air support game increases it's land based air arm to be much like Soviet Naval Aviation with heavy bombers and fighters to go with it's maritime patrol planes. The navy could also gain the army's coastal artillery role with a coastal defence force of fixed strong points, self propelled artillery and anti ship missiles and naval infantry.
Finally the USAF, which has lost frontal aviation to the army and strategic bombing to the USSWF, reinvents itself with an emphasis on air defence with SAM's and interceptors. After a few years this gets hived off into it's own arm.
So in 2014 you have something like this.
The Department of Defense which is subdivided into:
The Department of the Army: United States Army and United States Army Air Force
The Department of the Navy: United States Navy, United States Naval Aviation, United States Coastal Defense, United States Coast Guard and the United States Marine Corps
The Department of the Air Force: United States Air Force and United States Air Defense Force
The Department of Strategic Weapons: United States Strategic Weapons Force
The Department of Gendarmerie: United States Gendarmerie
The Department of Medical Services: United States Military Medical Corps