Just in the next 4 years following the upcoming 2016 season, the Cards are scheduled to play Purdue, Alabama, Notre Dame (twice), and Kentucky (4 times). So that makes an average of ten P5 games per season (2 non-ACC + 8 ACC). The Cards have room for one more P5 game in both 2019 and 2020.
Though I personally wish we would play an occasional opponent from the West, TJ has indicated that's not in his/our plans.
The closest conference that we've had the least number of games with is the Big Ten.
Given TJ's reluctance to play PAC12 and B12 games, and also the fact that the SEC doesn't ever want to play anyone credible out of conference (unless it's an arranged-by-TV game), I suspect we are relegated to hoping for an occasional Big Ten game.
If we can get it rolling again (string together several 10-2, 11-1 seasons), ESPN may come calling for another set-up game or two. I also think TJ enjoys using the extra money on his coaches, that you get when playing these high-payday neutral network-arranged games.