My original plan was to have a USAF Starfighter squadron have a deployment to the North of Vietnam similar to the Quemoy crisis in 1958. My thoughts are the Air Force originally selects the Starfighter, but after a bribery scandal emerges the position is reversed and the Mirage III selected.
This is for my Vietnamese TL, which is part of my signature block and for the Vietnamese Air Force in the mid 1960's.
If the PoD is around the late 1950s, it's possible to get the Super Tiger in service. I don't see the USAF going for the Mirage III. The United States government rarely adopts foreign designs and always Americanizes and license produces the few it does.
I'm not sure what role you are looking at having a Mirage III type aircraft fulfill, but I assume it's intended for multirole operations? The F-104 isn't really multirole though, more of a light interceptor/attack aircraft. If we're looking at aircraft with USAF origins (as opposed to USN aircraft) there are a few options. Republic proposed an interceptor variant of the F-105 to Canada, including the possibility of using the Orenda Iroquois. Convair proposed multirole F-106 variants to (IIRC) West Germany and Japan. The F-106 was a development of the F-102, which was used for attack missions in Vietnam. There's also the USN F-4, but the USAF was firmly behind the F-105 until McNamara made them procure F-4s instead.
As light aircraft go, there aren't too many USAF options. The Northrop F-5 was foreign aid for developing air forces and lacks radar and other features. Something like the F-20 might be possible, but at that point it's essentially a totally new aircraft due to the state of 1960s engine and avionics technology. It was only easy to implement modern equipment on a 1950s design because 1980s technology had developed so much. Once again, if the USAF is willing to go with a USN aircraft the Vought F-8 and Grumman F11F-1F Super Tiger are options. I think the Super Tiger is the best choice for a light fighter, but the USAF might have problems with it not originating within the USAF structure (that's part of what led to the failure of the F-20 in the 1980s, it was going up against the F-16 which had institutional support in the USAF).
If you can get the Super Tiger in service it (and/or variants of it) would likely still be in service to the present day. It started with the General Electric J79, so fuselage changes will be minimal if new engines are added later should it become part of a light fighter or new foreign aid fighter program in the 1970s.