Sigonella and Tripoli are roughly equal distance to Benghazi. This was a totally political decision not to involve the US military in a trans border operation.
There was a rapid response team that flew into Sigonella after another relief force from Tripoli was already on their way to and did relieve Benghazi along with CIA volunteers there.
They are what constituted the third firefight and extracted the remaining personnel except for Stevens who was found by friendlies.