Platt ("temperature") scaling works well, and it's very simple. Yes, you do it post-training, usually on a held-out calibration set. Some people will then retrain on all of the data and reuse the learned temperature, but that doesn't always work out as well as you want it to.

FTR, "multiclass classification" means each instance belongs to exactly one of many classes. When each label can be 0 / 1 irrespective of the other labels, it's referred to as "multilabel classification".