I think once you start running out of piles, you should be able to score on 3 or 4-card straights & flushes, but at a lesser value.
Example #1: You can only pick up 4 cards on a turn, but because it makes a 4-card run, it scores as a straight, minus 1 point.
Example #2: You can only pick up 3 cards, but they're of the same suit, so they score as a flush, minus 2 points.