1) There were no phases, it was just “show up, get poked” 2) There were plenty of vaccines to go around. Countries typically had millions of smallpox vaccine doses ready to go just in case 3) with no lockdowns, they could go to schools / churches / offices & line people up 4) the smallpox vaccine was incredibly heat-stable. There were batches that were still efficacious after being stored for a year at 113F 5) the public health infrastructure had a lot of practice with mass vaccinations (they did them all the time in other countries) And it did cause a city-wide panic. Smallpox has like a 20% fatality rate so people were pretty motivated to get the vaccine. All this is from this book https://www.amazon.com/Smallpox-Disease-Inside-Eradicating-Worldwide-ebook/dp/B002FU5O54/
A response from @politicalmath, based on Smallpox: The Death of a Disease by DA Henderson: