I had figured that having known most of the surviving winners for a long time, we was able to gauge their potential willingness to help well before ever arriving at the games. He is friends, and it sounds like fairly close friends, with a number of them, and so would be able to do this pretty well I think.
Additionally, and I can't really understand why they wouldn't surveil the tributes and entourage leading up to the games, they apparently don't keep them under watch or even listen to them via audio bugs.
This fact opens up at least some time for him to have met with and spoken to like minded winners (and a games master who for the life of me, how the heck did he get identified as a rebellion oriented capital citizen, I would really love to know). So far as coordination with District 13, it should not have taken much communication to coordinate. District 13 likely has been monitoring the goings on in Panem for a long time.
Seeing rebellion fomenting among the districts, and then getting even the most brief word through whatever means that a party of rebels is seeking asylum, respite, refit, whatever, I would accept them. Especially if I was a district that successfully managed to defend myself and survive against the Panem forces. There is little that Panem could do that would not expose the other districts to the fact that 13 exists still, and given their inability to destroy them previously, I doubt they would do so again when the bulk of their forces are currently putting down an internal rebellion.
Haymitches familiarity with the tributes let him coordinate with them quickly and out from under the less than watchful eye. District 13 did not really need a lot of information or time to process it to want to let them in.