I can report from this condominium garden apartment complex that the two steel trash/garbage bins near here were overflowing with bags of trash, garbage, discarded construction debris, and consumer items. At the time I predicted that rats would burrow into the garbage and that one rat gestation period later, we'd have baby rats in a flood. It hasn't happened yet.
However, my suggestion is as applicable to the Christmas trash rush as it is to a snow emergency. It's simple; prior to the holiday, or to the snow fall, while roads are still clear, residents would purge their homes of trash and garbage, in time for pickup by the contractor while roads were still clear. Then, during the holiday or emergency, residents would only place unsanitary garbage (diapers, food waste, cat litter, etc.) in the bins, and would *temporarily* store trash (boxes, paper, rinsed milk jugs) in their homes. Trash is very bulky compared to garbage.
I don't know how to achieve this resident behavior change. Maybe the Grim Reaper in costume could go 'round the neighborhood ringing his bell and calling out "Bring out our trash!" the way they brought out the dead during the Plague. The situation is complicated by our polyglot of many languages and cultures, but I think we can agree, can't we, that when times are tough, tough decisions (trash versus garbage) have to be made?