The Archons and Iotians. Earlier starships contacted them first. With the Capellans whatever ship McCoy was on contacted them first some years prior. I made a point of trying to exclude those. There are a few others including worlds where contact had obviously happened long before. I didn't include the Talosians because Pike got there first.
In fairness you have also to define what you're contacting. Is it sentient, intelligent, a race, a new species. A virus isn't an intelligent life form. The Horta and the two different cloud creatures were self-aware and intelligent life forms.
Strictly speaking a civilization goes beyond simply a new species or race unless the term civilization is being used very loosely and perhaps incorrectly. In Trek we often see representatives of intelligent life forms, but not their civilizations. So does that still count? If not then the instances of first contact are far fewer.