That's no longer a random sample ~ END OF STORY. That's when you must turn to other methods.
Yes, I get that this is your theory. The problem for you is your theory has been proven wrong. You only need look back to November to see that. There was more polling than ever before leading in to the 2012 election and it proved to be extremely accurate. You seem to be avoiding this fact. The state polling, in particular, was amazingly accurate - especially if you used an aggregate of the polls. I don't know how many times I have to say this. I think he's a snot nosed little liberal jerk, but Nate Silver (and others like him) who based their predictions on the polls got it right. In the case of 538 blog and others like it, they got all 50 states right. People like Dick Morris, Barone, etc, who rejected the polls got it wrong - and had to admit as much after the election.
They'll tell you how they dealt with the problem ~ that being the loss of randomnessin the process.
BTW, that's not a 'theory' ~ W. Edwards Deming had a corollary to the problem of declining responses or selections ~ that is, that the more samples you take over a long enough period of time, the closer you get to random selection ~ which means that the fellows with the month after month, week after week, day after day recurring samples of population where they asked the same question actually approached randomness ~ but not for a specific day, or a specific week, or even a specific month ~ those who took static highly stratified surveys could not possibly make enough sample selections to manage the flow of data. They'd never approach randomness in the process.
The point being that the only way to beat the small minority effect is to increase your total sample size into at least 11 times as you've ever planned on doing for a given situation.
If you were doing 1000 calls, you'd need to do 11,000 calls. If you had a multi-question, multi-issue sort of poll or survey, you could easily find yourself needing to do over 100,000 samples just to maintain randomness and to overcome the small minority effect.
Maybe some of them tried that ~ I've been listening closely and no one to my knowledge has come forward and volunteered such information.