FWIW, NYC has a right to shelter regime due to a court ruling some years ago, so they're technically entitled to somewhere in the shelter system. My dad used to work at an intake center so seeing immigrants show up seeking help wasn't new to me, but the difference is the numbers seeking shelter.

IIRC, hotels and temporary camps have been the main housing sites, but other alternatives have been abandoned Catholic schools and surplus land at a mental instutition. They floated using school gyms, but the parents revolted at the idea of cancelling gym for unvetted men with free access to the school. 

 Based on what I was seeing on my commute to work, with the 60 day cap on shelter stays, they're slowly starting to use the subway system as their overnight rest point. Most seem to find an outlying station, look for an empty corner, and setup some blankets for the night...