I applied through the portal, then let my departmental contact know the paperwork was filled out. I had one interview held by video call after initial introduction over email, then there were a couple of months of shuffling classes before getting a schedule.

The process was straightforward as I had previously been a Teaching Assistant and was a known quantity. They preferred to hire previous TAs over most other part-time applicants. I did have the implicit support of the Department Chair (who was ostensibly the manager for the position) and of the Calculus Coordinator, who developed, maintained, and guided the calculus curriculum and the teaching thereof for the department.