Connie,
I have no idea of the process our ancestors went through to register and eventually depart. I can only address the records that we have access to.
If you find your ancestor arrived in America on July 4, 1900, then go to the Azorean passaportes and start looking on July 4th, working your way back in time. I seem to remember Joao Ventura telling us in Salt Lake that the vast majority registered within 2 months of departure. I'm sure there are outliers and a few registered before that.
Most left from their home port. Ponta Delgada on Sao Miguel had those departing from S. Miguel and Santa Maria. This is the port that those going to Hawaii had to use. Then we have Angra on the island of Terceira. Mostly those from Terceira, probably Graciosa and some Sao Jorge island used Angra. Horta on Faial had a port. Pico and Faial used this port and possible some Sao Jorge. The records exist for these 3 ports with Horta's being fully in tact with no missing records. Ponta Delgada and Angra have a few gaps. The passaportes for the island Flores (and Corvo) are completely missing.
If an ancestor had to get to America right away, they could have left Pico on a smaller boat and gone over to Sao Miguel if that was the next boat going to America. They could use any port they wanted. The majority used their home port though, so start there.