If you accept Star Trek: The Animated Series you have the Kzin attacking Earth almost simultaneously with the Vulcans’ arrival in Montana. And if you accept the UNSS Icarus mission to Alpha Centauri, you have that ship making contact with the native Centaurans before the Vulcans pass though the Sol system.
Putting Star Trek: First Contact‘s World War III in the 2050s solves this problem neatly. The Icarus left Earth in 2048, with a ten-year flight plan reaches AlphaCent in 2058. Radio signals from AlphaCent take four-plus years, so it’s 2062 at the earliest when listening posts on Earth could detect the Icarus‘s signal. However, if Earth is trashed from World War III, there’s no one listening, or if someone is, the signal gets filed away.
In the 23rd and 24th centuries, there’s debate over which contact counts as the first. The Vulcan contact, or the Icarus contact. However, since the Vulcan contact seems to have had more butterfly effects, that’s the one that generally gets the nod.
The Kzin question I handle a little bit differently. Whatever propulsion system the Icarus uses, it would be the rough equivalent of a minor cosmic event, emitting tons of hard radiation and generally making a lot of noise in the background cosmic junk. That’s going to be pretty obvious to anyone who’s looking, and so I have the Kzin follow the Icarus into AlphaCent (relatively speaking; I put the Kzin there about the time the Icarus is ready to head back home). So, the Man-Kzin Wars Sulu and Spock talk about happened, but in the AlphaCent system.
Random, true. Competely non-canon. But it’s how I view the muddled years of the mid-21st century in the Star Trek universe, keeping all the cool stuff that’s been tossed around for years.