You don't need more clothes, you need fewer clothes, and wash them more often. I know it sounds backwards, but you won't have the overflowing situation, if you only have enough clothes, towels and blankets to fit the available storage, KWIM? I think everyone needs 10 days worth of undies and socks, 3 -5 pairs of pants and 10 shirts. Then maybe one coat each, 2-3 pairs of shoes, 3 sweaters or sweat shirts, and 3 pairs of pj's or pj substitute. 2-3 towels each. Put the rest in a big storage bin, and tell them they have to live with what they have for 3 weeks. You will do laundry on x days--towels on monday, sheets on Tuesday, etc. Whatever schedule you want. Once they learn to live with less, it will get easier.
Mom of 3, who learned this on vacation. We had less, we could live with less. I then bought less. Worked better allaround.