The William Tell Fighter Weapons & Air Defense meet competitions are famous from the early 50s.
During the operational period of the F-104 in the USAF also this aircraft participated in some of the meets and even won the 1962 episode (by 479TFW, George AFB).
Known participations include:
September 1959 - Air Defense Meet
Nr 337th FIS participated with a number of F-104A Starfighters.
Aircraft seen were: F-104A 56-847 and 56-876.
538th FIS based at Larson operating the F-104A has been reported to be there as well.
September 1962 - Fighter Weapons Meet
Nr 479th TFW from George AFB participated with a number of F-104C Starfighters and won the competition.
Aircraft seen were: F-104C 57-914, 57-915 and 57-916. Overall winning F-104C pilot was USAF Captain Charles "Chuck" E. Tofferi.
Beneath a photo taken at Nellis AFB of 57-914 during the Fighter Weapon Meet, taken on September 15th (USAF, thanks to Peter Mühlböck)
Nice video around this 1962 meet can be found here:
September 1963 - Air Defense Meet
Nr 151st FIS participated with a number of F-104A and at least one F-104B Starfighters.
Aircraft seen were: F-104A 56-823, 56-882 and F-104B 57-1302.
September 1965 - Air Defense Meet
Nr 319th FIS, based at Hamilton AFB participated with a number of F-104A Starfighters.
Capt J.D.Dunn (319FiS) was declared "William Tell Top Gun".
Likely also in 1960 the 479th TFW participated but we are checking for evidence.
See also video regarding this exercise (including some shots on 319FiS Starfighters, here: