Hiruzen Sarutobi is the third Hokage, while Minato is the fourth Hokage.
After the death of Minato, Hiruzen becomes Hokage again and he is still called the third.
But why still the third? Why isn't Hiruzen called the fifth then? And why didn't they choose a new candidate as new Hokage?
This is a guess from human behavior not exact knowledge.
they are used to call him the third Hokage and only know him as the thrid. Even if he now would officially be the fifth. Compare this to kings in the real world even if the went to exiles or somethign and came back to be kings again they would not change their name.
there might not be a suitable candidate at that point or he might be the best option.
After the kyubi attacked Konaha was basically a disaster zone .Hiruzen knew that if the other nations sense any weakness in the leadership of the village it could easily have led to another invasion.
That coupled with the fact that there simply was no one else that came close to Minto. Orochimaru had already defected. Jiriyaya spent most his time chasing after Orochimaru.There simply was no one else for it.
Sarutobi Hisuzen Sama never re-elected as the hokage of the konoha. Since he was the previous hokage, he was just represented as a hokage after the death of Namikaze Minato Sama(fourth).
Reasons were,
Everybody knew Danzo was not the right person for hokage post.
Jiraya sama was wandered around to find a pupil who was about to bring peace to shinobi world, and Ooruchimaru was very evil person and Tsunade Sama never accepted the post of hokage (This was until she met Naruto san).
Since he was just a representative, he was never called the fifth.
