It's not really important that the job runs once a second, but that it starts immediately when I want it to.
If I start a job with pg_cron, it will not be executed until the next full minute at the earliest.
Otherwise pg_cron with a function that performs a pg_sleep of one
second in a loop.
Anyway, it seems to me you are better refactoring your solution: it
seems you need to process data when _new data_ comes, not once per
second, so it sounds to me like a trigger could solve the problem.
The processing of the data via a job is deliberately chosen so as to separate the insertion of the data from their processing.
If a trigger were to do this, the transaction in which the data is inserted would take longer. This is not intended.
It is common for many records to be inserted in a short time, but processing takes a little time. The application that inserts the data should however not be slowed down.