It just seems like post TOS, unless using the exact means they used to get there (Borg tachyon hole, the wormhole in "Yesterday's Enterprise") then they're stuck.
Looking at Memory Alpha, this would seem to be the best answer: Ronald D. Moore
commented: "I would assume that the precise calculations involved in using the slingshot method are something of a closely-guarded secret.
So basically, probably only Spock knows how to do it. I can buy that.
Picard's line means that it can be done is probably public knowledge given there were hundreds of Enterprise crew, so it can't be "Captain's eyes only"
The fact that they were able to do it in a beat up looking Bird of Prey means its not "Enterprise only". I'm going with Moore here.